11th Grade Lesson Plan

Packet 4/27/2020-5/1/2020

 

Remote Learning Packet NB: Please keep all work produced this week. Details regarding how to turn in this work will be forthcoming.

April 27-May 1, 2020 Course: 11 Art Teacher(s): Ms. Frank clare.frank@greatheartsirving.org Weekly Plan: Monday, April 27 ⬜ Continue working on your project, with attention to craftsmanship in shading, mark-making and line quality. Tuesday, April 28 ⬜ Sketchbook entry: Area of strength and area for improvement ⬜ Continue working on your project Wednesday, April 29 ⬜ Sketchbook entry: Distinctive qualities of your project ⬜ Finish working on project Thursday, April 30 ⬜ Photography Project: Converging Lines and Sense of Place Friday, May 1 ⬜ Complete the formatting of your photography project. If you have time remaining, draw an image that represents your view onto the outside world, whether through a window, from a porch or balcony, or some other vantage point. Use any media you wish. Load a file of your image to the day’s entry. Devote 20 minutes of quality work time to your art assignments each day.

- The assignments will be submitted as a single PDF of photos at the end of the week, with the exception of the photography project, which will be submitted as a Google Slides document.

- For written sketchbook entries you have two options: a. To write them out in your sketchbook as we do in class normally, and include them

in the pdf upload at the end of the week b. To type them into a Google Doc assignment. This will be posted as an “ungraded”

assignment, but really it’s graded as part of the larger packet grade.

Monday, April 27 1. Continue working on your project, with attention to craftsmanship in shading, mark-making and line quality. Keep in mind your individual expressive intentions as you work. As you draw, remember to keep a clean folded piece of paper under your hand to avoid smudging. Tuesday, April 28 1. In a dated sketchbook entry, address an area of strength and an area for improvement in your project. 2. Continue working on your project, using your self-evaluation to help direct your priorities as you work. Wednesday, April 29 1. In a dated sketchbook entry, describe distinctive qualities of your project. What about the drawing style or composition speaks in a particular way of you, your touch, your interests, or your aesthetic judgment? There is a plethora of possibilities, and answering this question involves you looking thoughtfully and receptively at your own work. 2. Complete your project, looking at the image as a whole to achieve unity and harmony, to enhance visual interest, and to accentuate the individual qualities of your work with finishing touches. Thursday, April 30 1. Review the project guidelines for the photographic project, Converging Lines and Sense of Place, and view the demo example on Google Classroom. (Instead of displaying your photographs in a handmade book, you will display them in a Google Slides slideshow presentation.) 2. Proceed to photograph your chosen environment, taking at least 10 quality photographs, from which you will select at least 6 (per guidelines). Friday, May 1 1. Create a Google Slide document for “Converging Lines and Sense of Place” (ready for you on the assignment listing in Google Classroom). Create a title page and a body of 6-8 slides, followed by an end page, into which you could type a message, poem, short story or reflection relating to your project. Save into your GoogleDrive, and upload to the assignment on Google Classroom, then submit.

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Remote Learning Packet NB: Please keep all work produced this week. Details regarding how to turn in this work will be forthcoming.

April 27 - May 1, 2020 Course: 11 Calculus I Teacher(s): Mr. Simmons Weekly Plan: Monday, April 27 ⬜ Read my announcement on Google Classroom’s stream ⬜ Story time! ⬜ Read about suprema ⬜ Complete Problem 1 ⬜ Read the definition of an intersection Tuesday, April 28 ⬜ Read Ch. 11 of Spivak until p. 179 Wednesday, April 29 ⬜ Read Ch. 11, pp. 179-186 Thursday, April 30 ⬜ Read Ch. 11, pp. 186-189 ⬜ Optional proof (ungraded) Friday, May 1 ⬜ Complete Problem 11.6 (“Problem 6”)

Monday, April 27

1. If you have not done so already, please read the long announcement that I posted on the stream (at 8am today).

2. Story time! If it’s technologically feasible to do, please email me at least one sentence letting me know how you’re doing, and tell me a fun story. I miss you (yes, you).

As a way of reviewing and solidifying some of the material from the past few weeks, we are going to be reading Spivak. Some of this is review, so if anyone is bored, I challenge you to prove each of Spivak’s theorems yourself (before looking at Spivak’s proof), and then completing as many as you can of the very fascinating problems he has at the end of each chapter. You’ll enjoy them. In reading Spivak, we are coming in at Ch. 11. He will employ two terms that we have not covered, but which will not take you longer than today’s 40 minutes to become familiar with. In service of preparing us to read Chapter 11, spend today following these instructions:

3. Read about suprema (and infima) on pp. 116-18, starting at the beginning of the chapter and stopping at the statement of Theorem 7-1.

4. On a separate sheet of paper, complete Problem 8.1 (he calls it Problem 1, but to clarify which chapter it’s in, I’m calling it Problem 8.1) at the end of Ch. 8. (You will use this same piece of paper the whole week, turning in just this one item next Sunday.)

5. Read the very brief definition of “intersection” in the middle of p. 43 (in the paragraph that starts, “If f and g are any two functions…”).

Remember to read for understanding. That means possibly pausing on a single sentence and thinking about it for a while, maybe drawing a few diagrams to help you understand it. Don’t rush yourself, or you’ll simply struggle more later. Tuesday, April 28

1. Read Ch. 11, from the beginning of the chapter all the way through the proof of Rolle’s Theorem on p. 179. Read for understanding. (At times, Spivak will write something like, “I leave the remainder of this proof as an exercise for the reader.” Don’t feel obliged to complete those problems (but of course feel free to!).)

Though Spivak doesn’t label it as such, his Theorem 11.2 (he calls it Theorem 2, but to clarify which chapter it’s in, I’m calling it Theorem 11.2) is Fermat’s Theorem. I am aware that he has a slightly different definition of “critical point” from the one we learned. He only mentions points where f prime is zero, whereas we had previously included also points where f prime is undefined (so long as f itself was defined). We will be going with Spivak’s definition, as we will with everything from now on.

Wednesday, April 29

1. Read Ch. 11, until Theorem 11.5 (“Theorem 5”) on p. 186. (At times, Spivak will write something like, “I leave the remainder of this proof as an exercise for the reader.” Don’t feel obliged to complete those problems (but of course feel free to!).)

Thursday, April 30

1. Read Ch. 11, through p. 189. 2. (Optional, ungraded) Page 189 ends with a statement of L’Hopital’s Rule (LOE-pee-Tall’s). If you

would like to, try to prove it. I’m happy to read your proof and give you feedback. (If you’re going to prove it yourself, careful not to glance at spoilers on p. 190.)

Friday, May 1

1. Finish reading Ch. 11 (not including the appendix on convexity and concavity, unless you want to). (At times, Spivak will write something like, “I leave the remainder of this proof as an exercise for the reader.” Don’t feel obliged to complete those problems (but of course feel free to!).)

2. Answer Problem 11.6 (“Problem 6”), writing on Monday’s sheet of paper. 3. If you have extra time of the 40 minutes allotted for math today, spend it mastering the vocabulary

from Ch. 11

 

Remote Learning Packet NB: Please keep all work produced this week. Details regarding how to turn in this work will be forthcoming.

April 27 - May 1, 2020 Course : 11 Drama Teacher(s) : Mrs. Jimenez (margaret.cousino@greatheartsirving.org) Weekly Plan : Lines should be mastered by the end of this week! Monday, April 27 ⬜ Practice lines for 20 minutes Tuesday, April 28 ⬜ Practice lines for 20 minutes ⬜ 11am-12pm: Zoom Rehearsal - Act 1 (only actors with lines in Act 1 and Student Director) Wednesday, April 29 ⬜ Practice lines for 20 minutes Thursday, April 30 ⬜ Practice lines for 20 minutes Friday, May 1 ⬜ Practice lines for 20 minutes Statement of Academic Honesty I affirm that the work completed from the packet is mine and that I completed it independently. _______________________________________ Student Signature

I affirm that, to the best of my knowledge, my child completed this work independently _______________________________________ Parent Signature

Monday, April 27 - Practice lines for 20 minutes. Every day you should review the lines you have already mastered

without looking, then focus on a new scene. If you are a lead and have many lines, choose one scene to review and a new one to work on. Every day you should review a different scene you already have memorized and work for as many days as necessary to master the new one. Record your time on the sheet.

Line memorizing strategies: - Recite your lines OUT LOUD. Practice them like you will say them on stage – projecting,

appropriate speed and emotion, etc. Ask yourself why your character is saying what he/she says and that will help you interpret how to say the line.

- Run your lines with a friend or family member. They should read the lines of the other characters in your scenes while you practice your lines from memory.

- Practice your lines in front of a mirror—the bigger the better! Watch yourself—your facial expressions, how you move, stand, etc.—to be aware of how you look while saying your lines.

- Record yourself saying your lines and listen to the audio (even better if you record your cues!) - Write out your lines by hand (especially if you have a long speech, it is helpful to get it into your

memory through writing it out multiple times). - KNOW YOUR CUES! What line or action comes before you speak? - Run through the parts of the scenes in which you do not speak—what is your character doing

during those parts of the play? - After spending a period of time going over your lines, take a walk or a nap � - REMEMBER: Consistent practice is the key to success!

ALL LINES MUST BE MEMORIZED BY THE END OF THIS WEEK! Tuesday, April 28

- Practice lines for 20 minutes according to Monday’s directions. Record your time on the sheet. - If you are in Act 1 or the Student Director, join the Zoom meeting for online rehearsal from

11am-12pm. This hour will count as your rehearsal time for the week (unless you are woefully behind memorizing lines). Record that you attended the Zoom rehearsal in your sheet to turn in. If you’re not in Act 1, you don’t need to attend; just practice your lines like usual.

- Zoom link: https://zoom.us/j/94257807722?pwd=NDNHWFltclNkZnhCYS9QQzlPWDI3UT09

Wednesday, April 29 - Practice lines for 20 minutes according to Monday’s directions. Record your time on the sheet.

Thursday, April 30 - Practice lines for 20 minutes according to Monday’s directions. Record your time on the sheet.t.

Friday, May 1 - Practice lines for 20 minutes according to Monday’s directions. Record your time on the sheet.

Drama Weekly Line Memorization

Name: Week: 4/27-5/3

Day: Minutes practiced:

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

Minimum time: 20 mins/day, 5 days/week

I verify that this is a true and accurate account of the time I have spent memorizing my lines this past week.

Signature: Date:

 

Remote Learning Packet

April 27 - May 1, 2020 Course: 11 Greek Teacher(s): Miss Salinas annie.salinas@greatheartsirving.org Weekly Plan: Monday, April 27 ⬜ If you haven’t already, accept the invitation to join the Google Classroom for 11th Greek! ⬜ Worksheet: Counting; Exercise 8ι ⬜ 40 reps of exercise (20 cardinal, 20 ordinal) Tuesday, April 28 ⬜Worksheet: Exercise 8λ ⬜ 30 reps of exercise (cardinal) Wednesday, April 29 ⬜ Worksheet: 1st half of of ὁ Ὀδυσσευς και ὁ Αἰολος ⬜ 30 reps of exercise (ordinal) Thursday, April 30 ⬜ Worksheet: 2nd half of ὁ Ὀδυσσευς και ὁ Αἰολος ⬜ 40 reps of exercise (cardinal) Friday, May 1 ⬜ Come to Office Hours at 10:30am! (link available in the stream of our Google Classroom) ⬜ Worksheet: Exercise 8θ ⬜ 40 reps of exercise (ordinal)

A note on Google Classroom:

At time of writing this note on Wednesday, 22 April, just over half of the students in this class have

accepted the invitation to join the Google Classroom. Don’t languish in the dark! If you are having

trouble accessing it, please reach out to me or Ms. Weatherton, our registrar, who can help get you set up.

This week (Monday, April 27, when you are reading this), our work is completely posted on Google

Classroom in handy-dandy daily assignments you can complete without ever having to print and scan

anything, along with links to Friday’s Office Hours. If you are one of the nine students who has not

done so, please log in to your Google Classroom as soon as you have access to a computer and accept

the invitation to join 11th Greek!

A note on the second exercise (heh, literally) of each day:

This week, one of my favorite youtube channels that makes content related to teaching classical languages

(I’m aware of my critical level of nerdery) posted a video with tips on how to master counting in foreign

languages. What luck! His pro-tip is this: if you use the numbers of your target language to count reps of

exercise, your brain is much more likely to latch on to them. (Pending permission, I’ll link the video in

our Google Classroom. He’s better at explaining things than I am.) Our second assignment of each day

this week is this: do a number of reps of your choice of the following: push-ups, sit-ups, squats,

step-ups, calf raises, crunches, plans (in seconds), or wall-sits (in seconds) and count them in Greek.

If you need to make a paper copy of the numbers to hold or put in front of you for the first few days, do!

A note on sanity levels:

How are you? Are you staying sane? Are you either bored and unchallenged or overwhelmed? Are the

days of the week losing all meaning? Is time losing all meaning? Is meaning losing all meaning? Same. I

care about your sanity, especially as Greek relates to it. If you would like more of a challenge or if the

workload is getting to be too much, reach out and let me know. I’m creating these lessons week by

week, so I have the flexibility to adapt subsequent plans according to how the class is doing. So far it

seems alright, but if you have feedback, shoot it to me. Hey, you know a good way to send me feedback?

Logging on to your computer and emailing me! While you’re there, accept the invitation to join the

Google Classroom! (Okay, I’ll stop...I realize I may be a cause of your potentially low sanity levels.)

Monday, April 27

➔ Complete Monday’s one-page worksheet, which includes counting and Exercise 8ι.

➔ Today, do forty reps of your exercise of choice, counting twice to δεκα (cardinal numbers) and twice to δεκατος (ordinal numbers).

Tuesday, April 28

➔ Complete Tuesday’s one-page worksheet, which includes Exercise 8λ. Τhis worksheet is best done on Google Classroom.

➔ Today, do thirty reps of your exercise of choice, counting thrice to δεκα (cardinal numbers).

Wednesday, April 29

➔ Complete Wednesday’s two-page worksheet, which includes the first half of a story about Odysseus. Τhis worksheet is best done on Google Classroom.

➔ Today, do thirty reps of your exercise of choice, counting thrice to δεκατος (ordinal numbers).

Thursday, April 30

➔ Complete Thursday’s two-page worksheet, which includes the second half of a story about Odysseus. Τhis worksheet is best done on Google Classroom.

➔ Today, do forty reps of your exercise of choice, counting four times to δεκα (cardinal numbers).

Friday, May 1

➔ Come to Office Hours at 10:30am! Link available in the stream of our Google Classroom. Ask questions, give me feedback on your workload and the best format for worksheets, or at least connect on a human level with someone you’re not quarantined with.

➔ Complete Friday’s two-page worksheet, which includes Exercise 8θ. Τhis worksheet is best done on Google Classroom.

➔ Today, do forty reps of your exercise of choice, counting four times to δεκατος (ordinal numbers).

Monday

Counting to δεκα / Exercise 8ι Write the cardinal and ordinal numbers in Greek and English here.

CARDINAL

1.

2.

3.

4. τετταρες (m./f.), τετταρα (n.) - four

5.

6.

7.

8.

9.

10.

ORDINAL

1.

2.

3.

4. τεταρος, τεταρη, τεταρον - fourth

5.

6.

7.

8.

9.

10.

Exercise 8ι (from workbook)

Select the English equivalent from the word bank above and write it on the line in front of each Greek numerical adjective.

1. _______________ τεταρτος

2. _______________ ἑπτα

3. _______________ ἐνατος

4. _______________ ἑξ

5. _______________ δεκατος

6. _______________ ὀκτω

7. _______________ δευτερος

8. _______________ τρεις

9. _______________ δεκα

10. _______________ πεμπτος

11. _______________ ὀγδοος

12. _______________ πεντε

13. _______________ ἑν

14. _______________ τριτος

15. _______________ πρωτος

16. _______________ εἰς

17. _______________ δυο

18. _______________ ἑβδομος

19. _______________ ἑκτος

20. _______________ τετταρες

21. _______________ ἐννεα

22. _______________ μια

Tuesday

Exercise 8λ Τhis worksheet is best done on Google Classroom. On the space to the right of each sentence, translate the underlined phrase. Then, circle the type of time expression used.

1. We worked τρεις ἡμερας. ______________________

gen. of time within which dat. of time when acc. duration of time

2. It all took place τριῶν ἡμερῶν . ______________________

gen. of time within which dat. of time when acc. duration of time

3. She arrived τῇ τριτῇ ἡμερᾳ . ______________________

gen. of time within which dat. of time when acc. duration of time

4. They had the work finished πεντε ἡμερῶν . ______________________

gen. of time within which dat. of time when acc. duration of time

5. πασαν την ἡμεραν the farmer was working. ______________________

gen. of time within which dat. of time when acc. duration of time

6. The athlete received a prize τῇ πεμπῃ ἡμερᾳ . ______________________

gen. of time within which dat. of time when acc. duration of time

7. τῇ ὑστεραιᾳ they left for the harbor. ______________________

gen. of time within which dat. of time when acc. duration of time

8. They traveled πολυν χρονον. ______________________

gen. of time within which dat. of time when acc. duration of time

Wednesday

ὁ Ὀδυσσευς και ὁ Aἰολος - ἡμισυ πρωτον (“1st half”) This worksheet is best done on Google Classroom. Read the following passage and answer the comprehension questions.

Odysseus tells how he sailed on to the island of Aeolus, king of the winds, and almost reached home:

ἐπει δε ἐκ του ἀντρου ( the cave) του Κυκλωπος ἐκφευγομεν, ἐπανερχομεθα ( we return)

ταχεως προς τους ἑταιρους. οἱ δε, ἐπει ἡμας ὁρωσιν, χαιρουσιν. τῇ δ’ὑστεραιᾷ κελευω

αὐτους εἰς την ναυν αὐθις εἰσβαινειν. οὑτως οὐν ἀποπλεομεν.

1. What do Odysseus and his men do when they escape from the cave of the Cyclops?

_____________________________________________________________________________________

_____________________________________________________________________________________

2. What does Odysseus order his men to do the next day?

_____________________________________________________________________________________

_____________________________________________________________________________________

3. τῇ δ’ὑστεραιᾷ is which of the following?

gen. of time within which dat. of time when acc. duration of time

δι’ὀλιγου δε εἰς νησον Αἰολιαν ( Aeolia - the island of Aeolus, king of the winds)

ἀφικνουμεθα. ἐκει δε οἰκει ὁ Αἰολος, βασιλευς των ἀνεμων ( of the winds). ἡμας δε

εὐμενως ( kindly) δεχομενος πολυν χρονον ξενιζει (entertains). ἐπει δε ἐγω κελευω αὐτον

ἡμας ἀποπεμπειν, παρεχει μοι ἀσκον ( bag) τινα, εἰς ὅν ( which) παντας τους ἀνεμους

καταδει (he ties up) πλην (+gen, except) ἑνος, Ζεφυρου πραου ( gentle).

4. Where do Odysseus and his men arrive next?

_____________________________________________________________________________________

_____________________________________________________________________________________

5. How long do Odysseus and his men stay with Aeolus?

_____________________________________________________________________________________

_____________________________________________________________________________________

6. What does Aeolus give Odysseus at his departure?

_____________________________________________________________________________________

_____________________________________________________________________________________

7. What wind was not in the bag?

_____________________________________________________________________________________

_____________________________________________________________________________________

Thursday

ὁ Ὀδυσσευς και ὁ Aἰολος - ἡμισυ δευτερον (“2nd half”) This worksheet is best done on Google Classroom. Read the following passage and answer the comprehension questions.

ἐννεα μεν οὐν ἡμερας πλεομεν, τῇ δε δεκατῇ ὁρωμεν την πατριδα γην ( our fatherland).

ἐνταυθα δη ἐγω καθευδω. οἱ δε ἑταιροι, ἐπει ὁρωσι με καθευδοντα ( sleeping), οὑτω

λεγουσιν. “τι ἐν τῷ ἀσκῷ ( bag) ἐνεστιν; πολυς δηπου ( surely) χρυσος (gold) ἐνεστιν, πολυ

τε ἀργυριον ( silver), δωρα (gifts) του Αἰολου. ἀγετε δη ( come on!), λυετε τον ἀσκον και

τον χρυσον αἱρειτε ( take).”

1. How long do Odysseus and his men sail for?

_____________________________________________________________________________________

2. ἐννεα...ἡμερας is which of the following?

gen. of time within which dat. of time when acc. duration of time

3. τῇ δε δεκατῇ is which of the following?

gen. of time within which dat. of time when acc. duration of time

4. When they come within sight of their fatherland, what does Odysseus do?

_____________________________________________________________________________________

_____________________________________________________________________________________

5. What do his comrades think is in the bag?

_____________________________________________________________________________________

_____________________________________________________________________________________

ἐπει δε λυουσι τον ἀσκον, εὐθυς ( at once) ἐκπετονται ( fly out) παντες οἱ ἀνεμοι και

χειμωνα δεινον ποιουσι και την ναυν ἀπο της πατριδος γης ἀπελαυνουσιν ( they expel/push

away). ἐγω δε ἐγειρομαι ( wake myself up) και γιγνώσκω τι γιγνεται (happens). ἀθυμω ( I

despair) οὐν και βουλομαι ῥιπτειν ( to throw) ἐμαυτον ( myself) εἰς την θαλατταν. οἱ δε

ἑταιροι σῳζουσι με. οὑτως οὐν οἱ ἀνεμοι ἡμας εἰς την του Αἰολου νησον παλιν ( again)

φερουσιν.

6. What happens when the men open the bag?

_____________________________________________________________________________________

_____________________________________________________________________________________

7. How does Odysseus react when he wakes up?

_____________________________________________________________________________________

_____________________________________________________________________________________

8. Why doesn’t Odysseus carry out his plan of despair?

_____________________________________________________________________________________

_____________________________________________________________________________________

9. Where do the winds carry the ship at last?

_____________________________________________________________________________________

_____________________________________________________________________________________

Friday

Exercise 8θ

This worksheet is best done on Google Classroom. Answer the questions and translate the following sentences, which continue the story from Odysseus’ perspective.

1. ἐπει εἰς την νησον ἀφικνουμεθα, προς τον του Αἰολου οἰκον ἐρχομαι.

a. Which is correct about ἀφικνουμεθα?

2nd person sg: you arrive 1st person pl: we arrive 2nd person pl: y’all arrive

b. What case is του Αἰολου, and what keyword thus goes before it when translating?

gen: of dat: in, on acc: no keyword

Translation: _____________________________________________________________________

2. ὁ δε, ἐπει ὁρᾷ με, μαλα θαυμαζει και, “τι πασχεις;” φησιν, “τι αὐθις παρει;”

a. Are the verbs ὁρᾷ, θαυμαζει, and πασχεις in the active or middle voice?

active middle

b. Which punctuation mark is ; in Greek equivalent to in English?

- : ? ! , .

Translation: _____________________________________________________________________

3. ἐγω δε ἀποκρινομαι, “οἱ ἑταιροι αἰτιοι ( to blame) εἰσιν. τους γαρ ἀνεμους ( the

winds) ἐλυσαν ( they loosed). ἀλλα βοηθει ἑμιν, ὠ φιλε.”

a. Which of the following does NOT describe Odysseus’ tone?

suppliant/begging responsible/mature blaming/selfish

b. What little thing does Odysseus do in the last sentence to appeal to Aeolus and attempt to make him more willing to help? _____________________________________________________________________

Translation: _____________________________________________________________________

4. ὁ δε Αἰολος, “ἀπιτε (go away) ταχεως,” φησιν, “ἀπο της νησου. οὐ γαρ δυνατον

ἐστιν ὑμιν βοηθειν. οἱ γαρ θεοι δηπου (surely) μισουσιν (hate) ὑμας.”

a. Does Aeolus want to help Odysseus or not?

of course, yes! not at all

b. What mood is ἀπιτε?

indicative imperative infinitive subjunctive

Translation: _____________________________________________________________________

 

Remote Learning Packet NB: Please keep all work produced this week. Details regarding how to turn in this work will be forthcoming.

April 27 - May 1, 2020 Course: 11 Humane Letters Teacher(s): Mr. Brandolini david.brandolini@greatheartsirving.org

Mr. Mercer andrew.mercer@greatheartsirving.org Welcome to Week 5! In the hopes of providing the greatest opportunity to read Hamlet with the leisure and appreciation it deserves, we will be scaling back the written work of this week to primarily focus on enjoying this timeless cornerstone of the Western Tradition. Daily comments for each reading are included, kept deliberately brief to serve as signposting for major themes/moments to keep an eye out for before taking on the day’s reading. For this week, there will only be two reading quizzes, on Wednesday and Friday. We will revisit Hamlet for a deeper essay at the start of next week. Weekly Plan: Monday, April 27 ⬜ Read William Shakespeare, Hamlet, Act 1 (p. 3-69 of the Folger edition) Tuesday, April 28 ⬜ Read Hamlet, Act 2 (p. 73-119) Wednesday, April 29 ⬜ Complete Quiz on Hamlet, Acts 1&2 ⬜ Read Hamlet, Act 3 (p. 123-185) Thursday, April 30 ⬜ Read Hamlet, Act 4 (p. 189-235) Friday, May 1 ⬜ Read Hamlet, Act 5 (p. 239-287) ⬜ Complete Quiz on Hamlet, Acts 3-5 Statement of Academic Honesty I affirm that the work completed from the packet is mine and that I completed it independently. _______________________________________Student Signature

I affirm that, to the best of my knowledge, my child completed this work independently _______________________________________ Parent Signature

Monday, April 27 Read William Shakespeare, Hamlet, Act 1 (p. 3-69 of the Folger edition)

At long last, we come to the end of our series of tragedies! It is fitting that we should end with the greatest of all, Shakespeare’s Hamlet. If Sophocles’ Oedipus exemplifies the perfect distillation of Aristotle’s standards for tragic action, then Hamlet reinvigorates and elevates the movement to heights that have yet to be surpassed. In the most general sense, a tragic tale has three distinct parts:

1) a Fall that communicates the primary flaw, motive, or concern of the play; 2) a period of great stasis and suffering, or a Wandering; 3) a final Reconciliation, or a resolution of the play’s action that leaves the hero or the cosmos of the story in a greater order than it started. Thus far, we have seen that each play can tackle these stages to varying degrees of emphasis; in

Oresteia, each play essentially serves as an emphasis on each respective stage, while also containing a complete movement within each individual work. In Oedipus, the vast majority of emphasis is in the first aspect, the Fall of the king and its causes. Hamlet is somewhat unusual in that it places primary emphasis on the latter two, leaving much of the action of the Fall to before the start of the drama. A key reason for this is that Hamlet is written as much in a Christian tradition as it is in a literary one; by this I mean that just as Homer, Aeschylus, and Sophocles innately concerned with the traditions and cosmology of their society, so too is Hamlet directly informed by a Christian world: it will be especially important to read the play with an eye towards the redemptive aspect of the play.

For the time being, however, simply begin! Avoid the temptation to over-analyze or to write off the sufferings and actions of the characters as merely confused or selfish; either extreme will distract you from simply opening yourself to the play’s splendor. Tuesday, April 28 Read Hamlet, Act 2 (p. 73-119)

“Something is rotten in the state of Denmark” says Marcellus in Act 1.4. As we continue onward into Act 2, we should hopefully begin to see that this is very much apparent. One of the prevailing themes of the play is infection, sickness of soul, and sin, the “vicious mole of nature” in men since birth (1.4.26); according to the Ghost in 1.5, the very cause of the descent of Denmark is due to his death, wherein “sleeping in my orchard, / A serpent stung me. So the whole ear of Denmark / is by a forged process of my death / rankly abused.” (1.5.42-45). In this sense, the “Fall” of the drama is the Fall of Original Sin: Hamlet is plagued by the problem of sin and guilt, particularly as he is given the apparently impossible task of getting revenge for his father and remaining pure (much like Orestes’ mandate from Apollo). Pay attention to imagery of infection and gardens.

As we continue into Act 2, remember Hamlet’s plan to put on an “antic disposition”: pay special attention to his different moods and behaviors, namely when performing before others, when confiding in a single person, or when alone and engaging in a soliloquy. One of the play’s greatest ambiguities is to

what extent Hamlet’s antic disposition is an act. Hamlet very often discusses or emphasizes knowledge and ignorance: why is he so concerned with these, and do they indicate his tragic hamartia in any way? Wednesday, April 29 Complete Quiz on Hamlet, Acts 1&2 Read Hamlet, Act 3 (p. 123-185)

Hamlet’s famous reflection on the nature of suffering: what is it that paralyzes him so? Act 2 revealed that he has waited two months since the Ghost’s command. Notice that Hamlet does not ask whether “To live”, but rather “To be": his concern is existential, pondering whether man ought to exist at all, and is not merely born out of personal depression. His conversation with Ophelia following this soliloquy is also revealing of his assessment of human nature. Why does he speak to Ophelia the way he does? Is he trying to tell her something? How could these opening scenes be related to his concern over knowledge vs. ignorance?

Pay attention to how Hamlet reacts when approaching Claudius: we see both at their lowest and most desperate here, albeit in a somewhat ironic situation. As you continue reading, remember this moment and consider the consequences of Hamlet’s decision. His interaction with his mother, meanwhile, provides a stark contrast with our previous tragic heroes: Hamlet refuses to cross any “natural” boundaries when dealing with his mother, and instead seeks to help her by revealing the severity of her sin and the state of her soul to herself. Perhaps some good will come of Hamlet’s antic disposition? Thursday, April 30 Read Hamlet, Act 4 (p. 189-235)

Hamlet’s disposition has clearly changed--what has caused this? Does he feel any remorse or

regret for what he has done in Act 3? Has he become no better than Claudius himself? The state Hamlet leaves Denmark in under Claudius as he leaves for England is dire; Claudius will put the blame on Hamlet, but is this not also an indication of Claudius’ own inability to rule well? Notice also the stark contrast in the ways Laertes and Ophelia react to their father’s death. To what extent does Laertes’ thirst for revenge serve as a foil to Hamlet’s? Friday, May 1 Read Hamlet, Act 5 (p. 239-287) Complete Quiz on Hamlet, Acts 3-5

By the end, has Hamlet had any reconciliation, any change of heart? Like much of the play, it is ambiguous, but the most critical passage begins at line 5.2.233, where we see Hamlet answer the question posed earlier of whether or not “To be”. Is this moment a despairing admission of defeat, or a surrendering himself to God’s Providence? Think back to the theme of Denmark and Original Sin: has

Hamlet managed to reconcile his understanding of the problem of sin? The play leaves Denmark’s fate uncertain: has the rot been purged? Has Hamlet saved his family and his kingdom by revealing their errors? Once you’ve read, take some time to see if you can answer some of these questions for yourself. Next week, we will begin with a short essay that will attempt to assess these developments and see how they synthesize with the traditional, Aristotelian understanding of the tragic action.

__________________________________ __________________________________ Wednesday April 29 __________________________________ Reading Quiz: Hamlet, Acts 1&2 __________________________________ 1. What or who is it that Barnardo, Marcellus, and Horatio encounter during the night watch in scene 1?

A. a wild beast B. a royal ghost C. a celestial sign D. a fleeing criminal

2. Which of the three characters is initially doubtful of their encounter?

A. Horatio B. Barnardo C. Marcellus

3. What proposal does Horatio make at the end of scene 1?

A. to make known to Hamlet what they encountered in the night B. to make known to Hamlet Ophelia’s love for him C. to make known to Hamlet his uncle’s misdeeds D. to make known to Hamlet a plot against his life

4. Identify the speaker: “Do not forever with thy vailed lids / Seek for thy noble father in the dust. / Thou know’st ‘tis common; all that lives must die, / Passing through nature to eternity.”

A. King Claudius B. Polonius C. Hamlet D. Queen Gertrude

5. From where do Claudius and Gertrude try to persuade Hamlet from going?

A. London B. Norway C. Wittenburg D. Paris

6. With regard to what does Laertes tell Ophelia to exercise utmost caution?

A. Hamlet’s love toward her B. Claudius’s love toward her C. her father’s counsel D. her own judgement

7. Identify the speaker: “This above all: to thine own self be true, / And it must follow, as the night the day, / Thou canst not then be false to any man.”

A. King Claudius B. Laertes C. Polonius D. Horatio

8. What does the spirit of young Hamlet’s father reveal to him in the night?

A. that the kingdom of Denmark will be overrun by Norsemen B. that Polonius was plotting to have Laertes become the next king C. that Gertrude was not his real mother D. that Claudius murdered him to take the throne and the queen

9. What does Hamlet demand of Horatio and Marcellus at the end of the first act?

A. their allegiance to him in battle B. a vow of silence about what they saw that night C. money for him to travel to England D. their help solving a riddle

10. What does Ophelia report to her father in the opening scene of the second act?

A. Hamlet’s strange behavior toward her B. suspicious activity among Hamlet’s friends C. that she’s being courted by an English prince D. her ambition to become queen of Denmark

11. What does Claudius call on Rosencrantz and Guildenstern to do?

A. discover the source of treachery in the palace B. find out why Hamlet has changed and try to cheer him up C. assassinate the prince of Norway D. spy on Laertes in France

12. Identify the speaker: “O, what a rogue and peasant slave am I!”

A. Polonius B. Rosencrantz C. Guildenstern D. Hamlet

13. In two to three complete sentences, answer the following question: Does Hamlet have an optimistic or pessimistic view of the human condition? _____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

__________________________________ __________________________________ Friday May 1 __________________________________ Reading Quiz: Hamlet, Acts 3-5 __________________________________ 1. Identify the speaker: “And for your part, Ophelia, I do wish / That your good beauties be the happy cause / Of Hamlet’s wildness. So shall I hope your virtues / Will bring him to his wonted way again, / To both your honors.”

A. Polonius B. Queen Gertrude C. Laertes D. King Claudius

2. What does Hamlet make known to Ophelia in the first scene of Act 3?

A. that he was the cause of his father’s death B. that she was the cause of his father’s death C. that he once hated her but now loves her D. that he once loved her but no longer does

3. How does Claudius respond to what he sees of the play The Murder of Gonzago ?

A. he expresses remorse over his grave offense B. he shows pride in the fact that he was not caught in his misdeeds C. he gives a prize to the players for their skillful acting D. he banishes the players from the realm for their poor acting

4. Who does Hamlet strike dead before knowing his identity?

A. Claudius B. Polonius C. Laertes D. one of the players

5. Identify the speaker: “A man may fish with the worm that hath eat of a king and eat of the fish that hath fed of that worm.”

A. Polonius B. Claudius C. Rosencrantz D. Hamlet

6. To where does Claudius seek to send Hamlet?

A. England B. Norway C. Poland D. France

7. Whom does the “rabble” threaten to make king? A. Hamlet B. Horatio C. Laertes D. Polonius

8. Who captures Hamlet and sends him back to Denmark?

A. Fortinbras B. an unknown pirate C. English soldiers D. Polacks

9. What plan is hatched to ensure Hamlet is killed by “accident”?

A. put him to sea in a ship that isn’t seaworthy B. poison him in his sleep C. place a candle in an unsafe place near his quarters D. convince him to agree to a friendly bout of fencing with Laertes

10. Whose grave is being prepared at the beginning of Act 5?

A. Ophelia’s B. Polonius’s C. Hamlet’s D. Horatio’s

11. Identify the speaker: “I am justly killed with mine own treachery.”

A. Claudius B. Osric C. Hamlet D. Laertes

12. True or false: Hamlet and Laertes make amends in their final moments.

A. True B. False

13. In two to three sentences, answer the following question: What exactly is Hamlet asking when he poses the question, “To be or not to be” (3.1.64)? _____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

 

Remote Learning Packet

April 27-May 1, 2020 Course: 11 Precalculus Teacher(s): Mr. Simmons Weekly Plan: Monday, May 4 ⬜ Story time! ⬜ Read “Right triangles” ⬜ Problems 1-5 odd Tuesday, May 5 ⬜ Read “The Sine, Cosine, and Tangent functions” Wednesday, May 6 ⬜ Problems 1-5 (all) Thursday, May 7 ⬜ Read “Relationships between Trig Functions” Friday, May 8 ⬜ Problems 1-8 (all)

Monday, May 4

1. Story time! If technologically feasible, send me an email telling me a fun story that happened recently. I miss you! (yes, you)

Learning from packets is much harder than learning in class. I’ve found (after much searching) a good resource for learning trigonometry. We’ll be using it for the remainder of the year. It’s the second part of a book called Precalculus by Mr. Joseph Gerth. I’ve included it as a material on Google Classroom.

2. Read “Right triangles” from Gerth’s Precalculus (pp. 84-90). 3. Complete problems 1, 3, and 5 (pp. 90-91)

Tuesday, May 5

1. Read “The Sine, Cosine, and Tangent functions” (pp. 93-109). Wednesday, May 6

1. Complete problems 1-5 (pp. 109-110) on the same piece of paper from Monday. Thursday, May 7

1. Read “Relationships between the Trig functions” (pp. 113-121) Friday, May 8

1. Complete problems 1-8 (pp. 121-122) on the same piece of paper from Monday and Wednesday.

82

Part

two:

Tri

gono

met

ry

Trig

onom

etry

is th

e st

udy

of a

ngle

s an

d th

eir r

elat

ions

hip

to tr

iang

les.

As y

ou’ll

soo

n se

e,

we

atte

mpt

to

rela

te t

he a

ngle

s of

a t

riang

le t

o its

sid

es. I

t is

per

haps

uns

urpr

ising

tha

t th

is Pa

rt is

clo

sely

rel

ated

to

Geo

met

ry –

we’

ll as

k th

at f

amou

s qu

estio

n, “

Wha

t is

the

rela

tions

hip?

” W

hat

is in

tere

stin

g, h

owev

er,

is th

at t

his

Part

is

also

clo

sely

rel

ated

to

Alge

bra.

We

will

ofte

n w

ork

with

equ

atio

ns a

nd u

se a

lgeb

ra t

o ch

ange

the

m i

nto

som

ethi

ng d

iffer

ent.

This

reve

als o

ne re

ason

why

Trig

onom

etry

is st

udie

d: I

t is,

in so

me

sens

e, a

uni

on o

f tho

se

two

grea

t dis

cipl

ines

of G

eom

etry

and

Alg

ebra

. It t

akes

wha

t you

kno

w in

eac

h di

scip

line

and

appl

ies

it, s

o th

at y

ou c

an e

xpan

d ea

ch.

This

bran

ch o

f m

athe

mat

ics

is a

lso

stud

ied

at t

his

poin

t be

caus

e it

is a

ver

y im

port

ant

Calc

ulus

con

cept

. Cal

culu

s tea

cher

s and

text

book

s will

ass

ume

that

you

can

eva

luat

e, e

.g.,

sin𝜋 4 in

sec

onds

. You

will

also

be

expe

cted

to

have

dee

p in

sight

into

the

gra

phs

of t

he

Trig

onom

etric

func

tions

. Sin

ce y

ou’ll

ofte

n be

find

ing

the

area

of a

Trig

onom

etric

func

tion

or, p

erha

ps, d

eter

min

ing

its b

ehav

ior a

s it

goes

to in

finity

.

Man

y st

uden

ts s

trug

gle

with

Trig

onom

etry

. Thi

s is

mos

tly li

kely

due

to th

eir d

efic

ienc

ies

in G

eom

etry

, Alg

ebra

, or

both

. We

will

giv

e a

smal

l rev

iew

of

right

tria

ngle

Geo

met

ry,

whi

ch w

e th

ink

will

be

help

ful f

or y

ou. B

ut m

any

of y

ou m

ay n

eed

to d

o so

me

read

ing,

re

sear

ch, a

nd p

ract

ice

on y

our o

wn

to g

et y

ou u

p to

par

. Eve

n w

hen

you

are

wel

l-ve

rsed

in

the

pre-

requ

isite

s, yo

u m

ust a

lso b

e pr

epar

ed to

spe

nd ti

me

prac

ticin

g an

d m

aste

ring

new

tech

niqu

es a

s w

ell.

Anot

her

issue

tha

t yo

u m

ust

be a

war

e of

is t

hat

Trig

onom

etry

is v

ery

vert

ical

. Tha

t is

, an

ythi

ng le

arne

d se

rves

as

a fo

unda

tion

for t

he n

ext t

hing

. If y

ou d

on’t

lear

n so

met

hing

, th

eref

ore,

it’s

near

ly im

poss

ible

to p

rogr

ess.

The

corn

erst

one

of th

is Pa

rt is

mos

t ass

ured

ly

right

tria

ngle

s. Yo

u sim

ply

mus

t m

aste

r th

e sp

ecia

l rig

ht t

riang

les

pres

ente

d in

the

firs

t se

ctio

n. F

ailu

re to

do

so w

ill n

early

gua

rant

ee fa

ilure

. As

such

, you

mus

t app

roac

h ea

ch

sect

ion

with

due

dili

genc

e, a

nd m

aste

r any

and

all

of th

e co

nten

t.

§1

Lin

ear f

unct

ions

– 8

3

Uni

t fou

r Ri

ght t

riang

le T

rigon

omet

ry

Τριγωνομετρία

– G

reek

for

Tri

gon

omet

ry. L

iter

ally

mea

nin

g “t

rian

gle

mea

suri

ng.

”

Uni

t fou

r – 8

4

Trig

onom

etry

is a

bra

nch

of m

athe

mat

ics

that

look

s at

tria

ngle

s, an

gles

, and

circ

les.

It’s

a co

mpl

ex a

nd o

ften

rigor

ous

art

that

req

uire

s in

tuiti

on o

ver

skill

s. It

is i

mpe

rativ

e to

de

velo

p an

d m

aste

r th

e fo

unda

tiona

l kno

wle

dge

of T

rigon

omet

ry, b

ecau

se w

ithou

t it,

su

cces

s is

ver

y di

fficu

lt.

Alth

ough

we

assu

me

the

read

er k

now

s no

thin

g of

Trig

onom

etry

, it m

ight

be

wise

to h

ave

som

e ex

perie

nce

befo

re b

egin

ning

thi

s U

nit.

Addi

tiona

lly,

ther

e ar

e a

few

Geo

met

ry

theo

rem

s tha

t are

ass

umed

to b

e kn

own.

If a

ny o

f the

follo

win

g se

ems d

iffic

ult,

it is

high

ly

reco

mm

ende

d to

con

sult

the

prev

ious

text

s.

We

begi

n by

stu

dyin

g so

me

basic

Geo

met

ry.

Afte

r th

is, w

e th

en i

ntro

duce

the

bas

ic

Trig

onom

etric

func

tions

: Si

ne, c

osin

e, a

nd ta

ngen

t.

§1

Ri

ght t

riang

les

Sinc

e th

e w

ord

“Trig

onom

etry

” lite

rally

mea

ns “t

riang

le m

easu

re”,

perh

aps i

t isn

o su

rpris

e th

at w

e be

gin

wor

king

with

tria

ngle

s. Re

call

that

a tr

iang

le is

a p

olyg

on w

ith th

ree

side

s, an

d th

at a

rig

ht t

riang

le is

so

calle

d be

caus

e on

e of

its

angl

es m

easu

res

90

° (w

hich

we

defin

e as

a r

ight

ang

le).

Thus

, Fig

ure

21 is

a ri

ght t

riang

le.

This

simpl

e ob

ject

con

tain

s m

any

prof

ound

trut

hs, s

ome

of w

hich

you

may

hav

e le

arne

d in

Geo

met

ry c

lass

. Pe

rhap

s th

e m

ost

wel

l-kno

wn

and

prof

ound

tru

th i

s kn

own

as

Pyth

agor

as’ T

heor

em. I

t has

bee

n kn

own

and

used

for m

illen

nia,

con

tain

s m

ore

than

30

0

diffe

rent

pro

ofs,

and

is us

ed in

just

abo

ut e

very

bra

nch

of m

athe

mat

ics.

i

Pyth

agor

as’ T

heor

em

Giv

en a

rig

ht t

riang

le w

ith le

ngth

s 𝑎

,𝑏,𝑐

, whe

re 𝑐

is t

he h

ypot

enus

e an

d 𝑎

,𝑏 a

re t

he

legs

, the

n 𝑎

2+𝑏

2=𝑐

2.

i E

ven

stat

istic

s ha

s a

use

for P

ytha

gora

s’ Th

eore

m! Figu

re 2

1

All r

ight

ang

les

will

hav

e a

box

(like

the

one

next

to 𝐵

) to

deno

te th

e fa

ct th

at th

ey a

re 9

0°.

Reca

ll th

at it

is n

ot g

ood

enou

gh to

“loo

k lik

e” it

’s 9

0° -

itm

ust h

ave

the

box.

§1

Rig

ht tr

iang

les

– 85

We

will

usu

ally

pro

ve e

ach

basi

c an

d fo

unda

tiona

l ass

ertio

n in

this

cour

se. H

owev

er, w

e be

lieve

that

mos

t will

hav

e al

read

y pr

oved

this

theo

rem

on

thei

r ow

n. T

here

fore

we

will

no

t pro

ve th

is as

sert

ion

here

.ii

Exam

ple

1

Find

the

leng

th o

f the

miss

ing

side

in F

igur

e 22

.

The

hypo

tenu

se –

alw

ays

oppo

site

of t

he r

ight

ang

le –

is s

ide

leng

th 𝐴𝐵

, whi

ch h

as a

le

ngth

of 3

1. T

hus

we

can

say

that

𝑐, w

hich

rep

rese

nts

the

leng

th o

f the

hyp

oten

use

in

Pyth

agor

as’ T

heor

em, i

s 3

1. A

nd s

ince

𝐴𝐶

is o

ne o

f our

legs

, we

will

let

𝑎=

11

.

Giv

enth

is, w

e no

w h

ave

11

2+𝑏

2=

31

2

12

1+𝑏

2=

96

1.

This

is n

ow a

sim

ple

alge

bra

prob

lem

. Sol

ving

for 𝑏

we

get

𝑏2

=84

0

𝑏=2

8.9

8.

Som

e rig

ht tr

iang

les

are

spec

ial;

they

com

e in

kno

wab

le ra

tios.

And

if w

e kn

ow th

e ra

tios

of a

ll th

ree

sides

, the

n w

e do

not

nee

d to

use

Pyt

hago

ras’

Theo

rem

at a

ll.

Cons

ider

the

right

tria

ngle

sho

wn

in F

igur

e 23

.

ii T

he in

tern

et is

a g

reat

reso

urce

. Use

it.

Figu

re 2

2

Pict

ures

are

not

(and

nev

er w

ill b

e) d

raw

n pe

rfect

ly to

sca

le.

Uni

t fou

r – 8

6

This

is a

n is

osce

les

tria

ngle

, whi

ch m

eans

tha

t tw

o of

the

sid

e le

ngth

s ar

e co

ngru

ent.

Cons

eque

ntly

, we

can

mak

e th

e fo

llow

ing

asse

rtio

n.

Isos

cele

s rig

ht tr

iang

le s

ide

ratio

G

iven

an

isosc

eles

righ

t tria

ngle

with

a le

g of

leng

th𝑠,

then

the

othe

r leg

has

a le

ngth

of

𝑠 a

nd th

e hy

pote

nuse

has

a le

ngth

of 𝑠√2.

Proo

f.

Cons

truc

t iso

scel

es ri

ght t

riang

le 𝐴𝐵𝐶

with

leg 𝐴𝐵

=𝑠.

The

n 𝐴𝐶

=𝑠

beca

use

we

have

an

isosc

eles

righ

t tria

ngle

.

Furt

her,

we

know

that

the

leng

th o

f the

hyp

oten

use

is

𝐴𝐵

2+𝐴𝐶

2=𝐵𝐶

2

by P

ytha

gora

s’ Th

eore

m.

But

we

know

the

len

gth

of b

oth 𝐴𝐵

and

𝐴𝐶

, an

d so

, by

su

bstit

utio

n, w

e ha

ve

𝑠2

+𝑠

2=𝐵𝐶

2.

Whe

nce,

afte

r sim

plifi

catio

n, w

e ha

ve

2𝑠

2=𝐵𝐶

2

𝐵𝐶

=𝑠 √2

,

Whi

ch is

wha

t we

wan

ted

to s

how

.

Figu

re 2

3

The

tick

mar

ks o

n th

e tw

o le

gs in

dica

te th

at th

ose

two

lines

are

the

sam

e le

ngth

.

§1

Rig

ht tr

iang

les

– 87

Exam

ple

2

Det

erm

ine

the

leng

ths

of th

e m

issin

g sid

es in

Fig

ure

24.

Appl

ying

the

spec

ial t

riang

le ra

tio, w

e ha

ve th

at 𝑠

=1

5. T

hus

the

othe

r leg

,

𝑌𝑍

=1

5,

and

the

hypo

tenu

se,

𝑋𝑍

=1

5√2

.

Keep

in m

ind

Pyth

agor

as’s

Theo

rem

can

stil

l be

used

to d

eter

min

e th

e pr

evio

us tr

iang

le,

alth

ough

you

wou

ld h

ave

to a

pply

the

fact

that

𝑋𝑌

=𝑌𝑍

.

Let’s

take

a lo

ok a

t one

mor

e sp

ecia

l rig

ht tr

iang

le ra

tio.

30

°−

60

°−

90

° Rig

ht tr

iang

le ra

tio

Giv

en a

righ

t tria

ngle

with

ang

les

of 3

0°,

60

°, a

nd 9

0°,

then

the

shor

t leg

has

leng

th 𝑠

, th

e lo

nger

leg

has

a le

ngth

of 𝑠√

3, a

nd th

e hy

pote

nuse

has

a le

ngth

of 2𝑠.

W

e w

ill p

rove

the

prev

ious

resu

lt, b

ut o

ur p

roof

will

be

a bi

t mes

sy. W

e’ll

look

for a

bet

ter

way

to p

rove

it a

s w

e co

ntin

ue th

roug

h th

is U

nit.

Proo

f.

To fu

lly p

rove

this

asse

rtio

n, w

e w

ill n

eed

to p

rove

thre

e di

ffere

nt p

ossib

ilitie

s:

Figu

re 2

4

Uni

t fou

r – 8

8

I. A

right

tria

ngle

with

legs

of 𝑠

and

𝑠√

3,

II.

A rig

ht tr

iang

le w

ith a

leg

of 𝑠

and

a h

ypot

enus

e of

2𝑠,

and

III

. A

right

tria

ngle

with

a le

g of

𝑠√

3 a

nd a

hyp

oten

use

of 2𝑠.

We

will

pro

ve th

e fir

st p

oint

, the

n le

ave

the

final

two

proo

fs to

the

read

er.

We

are

give

n a

right

tria

ngle

with

leng

ths

of 𝑠

and

𝑠√

3. T

hus,

usin

g Py

thag

oras

, we

have

𝑎=𝑠,𝑏

=𝑠 √

3

whe

nce

𝑠2

+(𝑠√

3)2

=𝑐

2.

Cons

eque

ntly

, we

get t

hat

𝑠2

+3𝑠

2=𝑐

2

4𝑠

2=𝑐

2

𝑐=2𝑠,

whi

ch is

wha

t we

wan

ted

to s

how

.

Exam

ple

3

Com

plet

e th

e rig

ht tr

iang

le s

how

n in

Fig

ure

25.

To “

com

plet

e” a

rig

ht t

riang

le m

eans

to

dete

rmin

e al

l of

its

sid

e le

ngth

s an

d an

gles

m

easu

res.

In o

ur c

ase,

we

mus

t det

erm

ine

one

angl

e m

easu

re a

nd th

e le

ngth

of b

oth

legs

. W

e be

gin

with

the

angl

e m

easu

re. R

ecal

l fro

m G

eom

etry

that

the

sum

of t

hree

ang

les

in

a tr

iang

le is

alw

ays

18

0°.

Thus

our

miss

ing

angl

e (w

hich

we’

ll ca

ll 𝛼

)iii is

𝛼=

18

0°−( 3

0°−

90

°)

iii It

is m

y cu

stom

to n

ame

side

s w

ith lo

wer

-cas

e La

tin le

tter

s, 𝑎

,𝑏,𝑐

, and

then

cal

l the

ir op

posit

e an

gles

w

ith lo

wer

-cas

e G

reek

lett

ers, 𝛼

,𝛽,𝛾

, res

pect

ivel

y. T

his

is m

y ow

n pe

rson

al c

onve

ntio

n, a

nd I

wel

com

e yo

u to

dev

elop

you

r ow

n.

Figu

re 2

5

§1

Rig

ht tr

iang

les

– 89

𝛼=

60

°.

We’

ll us

e th

e ra

tios

show

n in

our

pre

viou

s re

sult

to d

eter

min

e th

e sid

e le

ngth

s. Re

call

from

Geo

met

ry th

at th

at th

e sh

orte

st si

de o

f a tr

iang

le w

ill a

lway

s be

oppo

site

the

shor

test

an

gle.

Sin

ce, i

n a

30

°−

60

°−

90

° tr

iang

le, t

he h

ypot

enus

e is

alw

ays

twic

e th

e le

ngth

of

the

shor

test

sid

e (w

hich

we’

ll ca

ll 𝑏

), w

e ca

n w

rite

14

=2𝑏

,

And

henc

e

𝑏=

7.

To d

eter

min

e th

e on

e re

mai

ning

sid

e, w

hich

is th

e lo

nger

leg

(sin

ce it

is o

ppos

ite th

e 6

0°

angl

e), w

e si

mpl

y m

ultip

ly o

ur p

revi

ous

resu

lt by

√3

. Hen

ce o

ur r

emai

ning

sid

e (w

hich

w

e’ll

call 𝑎

) is

𝑎=

7√

3.

We

show

our

fina

l res

ults

in F

igur

e 26

.

Pyth

agor

as’ T

heor

em a

llow

s us

to d

eter

min

e th

e th

ird s

ide

of a

righ

t tria

ngle

whe

n on

ly

give

n tw

o ot

her

sides

. Thi

s is

very

use

ful,

but

it le

aves

out

som

e ot

her

info

rmat

ion.

For

ex

ampl

e, w

hat

if w

e w

ante

d to

det

erm

ine

the

angl

e m

easu

res

whe

n gi

ven

two

side

leng

ths?

Unl

ess w

e m

easu

re it

, we

wou

ldn’

t be

able

to te

ll.iv O

r, w

hat i

f we

had

only

kno

wn

one

of th

e sid

es o

f our

tria

ngle

? Th

en w

e co

uld

not d

eter

min

e an

ythi

ng e

lse.

This

is o

ne o

f the

rea

sons

tha

t th

ese

spec

ial r

ight

tria

ngle

s ar

e so

use

ful.

We’

ll us

e th

at

fact

a m

yria

d of

tim

es in

this

Part

– a

nd w

e’ll

com

e ba

ck to

it v

ery

shor

tly. D

oubt

less

, of

cour

se, y

ou a

lso s

ee th

e ad

vant

age

to s

olvi

ng a

righ

t tria

ngle

as

quic

kly

and

effic

ient

ly a

s w

e ju

st d

emon

stra

ted.

iv A

nd e

ven

if w

e di

d m

easu

re it

, we

wou

ld h

ave

to d

raw

our

tria

ngle

per

fect

ly to

sca

le, a

nd e

ven

then

, th

ere

is a

deg

ree

of e

rror

in m

easu

rem

ent.

Figu

re 2

6

Uni

t fou

r – 9

0

Allo

w u

s to

reite

rate

the

impo

rtan

ce o

f thi

s se

ctio

n: Y

ou s

impl

y m

ust b

ecom

e ve

ry a

dept

at

sol

ving

spe

cial

righ

t tria

ngle

s. W

e ca

nnot

ove

rem

phas

ize

how

foun

datio

nal t

his

is. Y

ou

will

util

ize

this

tool

aga

in a

nd a

gain

in th

e fo

rthc

omin

g se

ctio

ns. W

e’ve

giv

en y

ou m

any

prac

tice

prob

lem

s in

thi

s se

ctio

n, b

ut y

ou w

ill n

eed

to t

ake

extr

a ca

re t

o m

aste

r it

for

your

self.

§𝟏 E

xerc

ises

1.)

In th

e fo

llow

ing

prob

lem

s, as

sum

e th

at 𝑎

and

𝑏 a

re th

e le

ngth

s of

legs

, and

that

𝑐

is th

e le

ngth

of a

hyp

oten

use.

Fin

d th

e le

ngth

of t

he m

issin

g sid

e.

(A) 𝑎

=12

,𝑏=

15

(B

) 𝑎

=1

.5,𝑏

=2

.3

(C) 𝑎

=24

,𝑏=

8

(D) 𝑎

=12

.5,𝑐

=1

9

(E)

𝑏=

11

3,𝑐

=2

01

(F

) 𝑐

=4

.5,𝑎

=1

(G

) 𝑐

=2

7,𝑏

=24

(H

) 𝑎

=1

0,𝑏

=1

2.

) As

sum

e yo

u ha

ve a

tria

ngle

with

leg

leng

ths

of 5

and

7.

(A)

Troy

lets

𝑎=

5 a

nd 𝑏

=7

. Tin

a le

ts 𝑎

=7

and

𝑏=

5. D

o th

ey g

et th

e sa

me

resu

lt fo

r 𝑐?

(B)

Prov

e th

at 𝑎

and

𝑏 a

re in

terc

hang

eabl

e w

hen

usin

g Py

thag

oras

’ The

orem

. 3.

) Ar

e al

l tria

ngle

s po

ssib

le?

For

exam

ple,

can

you

hav

e a

right

tria

ngle

with

legs

of

leng

th 1

0 a

nd 2

0 a

nd a

hyp

oten

use

of 1

5?

Why

or w

hy n

ot?

4.)

Reca

ll th

at a

ll tr

iang

les’

angl

es a

dd u

p to

18

0°.

With

thi

s in

min

d, c

ompl

ete

the

follo

win

g rig

ht tr

iang

les.

(A)

(B)

(C)

(D)

5.

) So

far,

we’

ve s

een

that

𝑎2

+𝑏

2=𝑐

2 is

tru

e fo

r al

l rea

l num

bers

– t

hat

is, a

t le

ast

one

of t

he t

hree

len

gths

was

irr

atio

nal.

Is i

t po

ssib

le t

hat

all

thre

e le

ngth

s ar

e in

tege

rs?

Let’s

exp

lore

…

(A) 𝑎

=3

,𝑏=4

,𝑐=

?

§1

Rig

ht tr

iang

les

– 91

(B)

𝑎=

?,𝑏

=12

,𝑐=

13

(C

) Se

eif

you

can

disc

over

thre

e ot

her P

ytha

gore

anTr

iple

s.6.

) Th

e fo

llow

ing

tabl

e re

pres

ents

a s

mal

l lis

t of

Pyt

hago

rean

Trip

les.

(Eac

h ro

w

cont

ains

the

thre

e sid

e le

ngth

s of

the

tria

ngle

.) Si

de le

ngth

s 3

4

5

Si

de le

ngth

s 6

8

1

0

Side

leng

ths

9

12

1

5

(A)

Wha

t pat

tern

exi

sts

in th

is ta

ble?

(B

) U

se th

e pa

tter

n to

list

out

thre

e m

ore

Pyth

agor

ean

Trip

les.

(C)

How

man

y Py

thag

orea

n Tr

iple

s ex

ist?

(D)

Thes

e w

ere

all m

ultip

les

of t

he s

mal

lest

(and

mos

t co

mm

on) P

ytha

gore

an

Trip

le, t

he 3

,4,5

tria

ngle

.v Will

this

tric

k w

ork

with

oth

er d

istin

ct P

ytha

gore

an

Trip

les,

like

the

5,12

,13

tria

ngle

? 7.

) Co

mpl

ete

the

follo

win

g rig

ht tr

iang

les.

(A)

(B)

(C)

(F)

(G)

v W

e hi

ghly

enc

oura

ge y

ou to

mem

oriz

e th

e 3

,4,5

tria

ngle

. It p

ops

up e

noug

h th

at m

emor

izin

g it

will

sav

e yo

u so

me

cons

ider

able

tim

e.

Uni

t fou

r – 9

2 (D)

(E)

(H)

8.)

As a

mat

ter

of c

onve

ntio

n, w

e of

ten

rew

rite

any

ratio

s th

at h

ave

an i

rrat

iona

l nu

mbe

r in

the

ir de

nom

inat

or s

uch

that

the

num

erat

or i

s en

tirel

y ra

tiona

l. Fo

r ex

ampl

e,

1 √2

=1 √2∙√2

√2

=√2 2

.

In m

ost c

ases

, thi

s is

pur

ely

by c

onve

ntio

n. O

ne re

ason

for d

oing

this

is th

at y

ou

will

onl

y ne

ed to

mem

oriz

e on

e se

t of n

umbe

rs w

hen

it co

mes

to s

peci

fic T

rig

ratio

s.

(A)

Wha

t is

the

den

omin

ator

in t

he e

xam

ple

show

n ab

ove?

And

wha

t is

the

nu

mer

ator

and

den

omin

ator

of t

he n

umbe

r mul

tiplie

d by

1 √2?

(B)

Wha

t is √

2

√2?

(C)

Base

d on

you

r an

swer

to

(B),

has

the

valu

e of

1 √2 c

hang

ed a

fter

the

mul

tiplic

atio

n? U

se a

cal

cula

tor i

f you

’re n

ot c

onvi

nced

. (D

) U

se th

e ab

ove

exam

ple

to ra

tiona

lize

the

follo

win

g de

nom

inat

ors.

i. 3 √3

ii.

4 √2

iii.

√2

√3

iv.

𝜋 √7

v.

3 √4

vi.

5

2√

2

vii.

24

√1

2

viii.

√3

√10

9.)

Answ

er T

rue

or F

alse

. (A

) A 4

5°−4

5°−

90

° tr

iang

le c

an n

ever

hav

e th

ree

side

leng

ths

whi

ch a

re a

ll w

hole

num

bers

.

§1

Rig

ht tr

iang

les

– 93

(B)

The

shor

test

sid

e of

a tr

iang

le is

alw

ays

oppo

site

of th

e sh

orte

st a

ngle

. (C

) Th

e hy

pote

nuse

is a

lway

s th

e lo

nges

t sid

e of

a tr

iang

le.

(D)

A tr

iang

le c

an n

ever

hav

e m

ore

than

one

90

° ang

le.

(E)

Ther

e is

an

infin

ite a

mou

nt o

f Pyt

hago

rean

Trip

les.

10.)

In y

our o

wn

wor

ds, d

escr

ibe

the

shor

tcut

to c

ompl

etin

g an

isos

cele

s rig

ht tr

iang

le.

11.)

In y

our o

wn

wor

ds, d

escr

ibe

the

shor

tcut

to c

ompl

etin

g a

30

°−

60

°−

90

° tria

ngle

. 12

.) W

ill y

ou fo

rget

thes

e sp

ecia

l tria

ngle

ratio

s?

13.)

Whe

re w

ill y

ou lo

ok if

you

forg

et th

em?

14.)

You’

re s

ure

you

won

’t fo

rget

them

??

§2

Th

e Si

ne, C

osin

e, a

nd T

ange

nt fu

nctio

ns

We

used

Geo

met

ry in

the

pre

viou

s se

ctio

n (a

nd w

e w

ill fr

eque

ntly

fall

back

on

it) a

nd it

ha

d so

me

good

use

s, bu

t in

this

sect

ion

we

will

dis

cove

r a n

ew s

et o

f too

ls th

at w

ill a

llow

us

to d

o ev

en m

ore.

Mor

e im

port

antly

, tho

ugh,

is th

at w

e’ll

be a

ble

to re

late

ang

le m

easu

res

and

side

leng

ths.

Cert

ainl

y, t

his

will

allo

w u

s to

cal

cula

te m

ore

than

we

coul

d pr

evio

usly

, but

we

will

also

fin

d a

shre

d of

trut

h th

at w

e fe

lt w

as th

ere

but c

ould

n’t q

uite

asc

erta

in.

To b

egin

with

, we’

ll de

fine

a fu

nctio

n th

at ta

kes

an a

cute

ang

le a

s its

inpu

t and

retu

rns

a ra

tio o

f tw

o sid

e le

ngth

s.

The

Sine

func

tion

Th

e Si

ne fu

nctio

n ac

cept

s as a

n in

put a

n ac

ute

angl

e of

a ri

ght t

riang

le, a

nd th

en re

turn

s th

e ra

tio o

f the

sid

e op

posit

e to

the

angl

e to

the

hypo

tenu

se. S

ymbo

lical

ly,

sin𝛼

=𝑎 𝑐

.

We

coul

d al

so s

tate

that

sin𝛽

=𝑏 𝑐

.

Not

e th

at o

ur c

urre

nt d

efin

ition

onl

y ac

cept

s ac

ute

angl

es o

f a ri

ght t

riang

le.

Uni

t fou

r – 9

4

Exam

ple

1a

Writ

e th

e ra

tio o

f the

sin

e of

ang

le 𝛼

and

𝛽 g

iven

Fig

ure

27.

Let’s

sta

rt w

ith th

e sin

e of

ang

le 𝛼

, whi

ch is

den

oted

as

sin𝛼

. Thi

s is

just

a ra

tio –

so

all w

e ne

ed to

do

is cr

eate

a fr

actio

n. S

ince

we’

ve d

efin

ed th

is fu

nctio

n as

the

side

oppo

site

to

the

hypo

tenu

se, w

e ju

st n

eed

to w

rite

sin𝛼

=3 6.7

,

Sinc

e 3

is th

e sid

e op

posit

e an

d 6

.7 is

the

hypo

tenu

se. I

t rea

lly is

that

sim

ple!

The

sine

of a

ngle

𝛽 is

han

dled

the

sam

e w

ay, e

xcep

t no

tice

that

the

sid

e op

posit

e 𝛽

is

diffe

rent

than

the

side

oppo

site

of 𝛼

. In

this

case

we

have

sin𝛽

=6 6.7

.

Wha

teve

r ang

le 𝛼

is, w

hen

we

inpu

t it i

nto

the

sine

func

tion,

we

rece

ive

as a

n ou

tput

3 6.7

.

The

sam

e is

true

for 𝛽

; whe

n w

e in

put 𝛽

into

the

sine

func

tion,

we

rece

ive

as a

n ou

tput

6 6.7

.

Of c

ours

e, th

at h

asn’

t rea

lly re

veal

ed to

us

any

new

info

rmat

ion.

Let

’s tr

y a

prob

lem

that

te

ache

s us

som

ethi

ng, s

hall

we?

Exam

ple

1b

Eval

uate

sin

30

° giv

en F

igur

e 28

.

Figu

re 2

7

Figu

re 2

8

§2

The

Sin

e, C

osin

e, a

nd T

ange

nt fu

nctio

ns –

95

To a

nsw

er th

is, w

e sim

ply

need

to s

et u

p ou

r rat

io. B

ased

on

wha

t we’

ve b

een

give

n, w

e ha

ve

sin

30

°=

2

hyp

ote

nu

se .

But

wha

t is

the

leng

th o

f th

ehy

pote

nuse

? W

e ca

n ea

sily

find

that

out

usi

ng w

hat

we

lear

ned

in th

e pr

evio

us se

ctio

n.i S

ince

2 is

the

leng

th o

f the

shor

t leg

, the

n th

e hy

pote

nuse

m

ust

be t

wic

e th

at le

ngth

. Hen

ce w

e co

nclu

de t

hat

they

hyp

oten

use

has

a le

ngth

of 4

. Th

eref

ore

sin

30

°=2 4

=1 2

.

So w

hat d

id w

e fin

d ou

t? W

ell,

if yo

u in

put 3

0° i

nto

the

sine

func

tion,

you

sho

uld

get o

ut

1 2. B

ut th

ere’

s a

very

impo

rtan

t que

stio

n to

ans

wer

: W

ill th

is on

ly h

appe

n w

hen

the

shor

t

leg

of a

30

°−

60

°−

90

° tr

iang

le h

as a

leng

th o

f 2?

Put a

noth

er w

ay, d

oes

the

size

of t

he

tria

ngle

pla

y a

role

in d

eter

min

ing

the

resu

lt of

sin

30

°?

To a

nsw

er t

his

ques

tion,

we

will

do

as a

ll m

athe

mat

icia

ns d

o:

We

will

exp

lore

, we

will

ex

perim

ent,

we

will

pla

y! W

e’ll

mak

e a

new

tria

ngle

with

an

angl

e of

30

° tha

t has

a d

iffer

ent

size

and

obse

rve

wha

t hap

pens

.

Exam

ple

1c

Eval

uate

sin

30

° usi

ng F

igur

e 29

.

i T

his

is e

xact

ly w

hat w

e m

ean

we

say

that

you

will

nee

d to

be

very

ade

pt a

t the

pre

viou

s se

ctio

n. T

his

is

ofte

n ho

w w

e’ll

calc

ulat

e ou

r Trig

func

tions

– b

y se

ttin

g up

a s

peci

al ri

ght t

riang

le, a

nd fi

lling

in th

e re

st o

f th

e in

form

atio

n. T

here

fore

, we’

ll st

ate

it ag

ain:

Lea

rn th

e pr

evio

us s

ectio

n!

Figu

re 2

9

Uni

t fou

r – 9

6

We

appl

y th

e sa

me

proc

edur

e. In

thi

s ca

se, t

he h

ypot

enus

e m

ust

be 2

0. S

ince

the

sin

e ra

tio is

opposite

hypotenuse

, and

the

side

leng

th o

ppos

ite to

30

° is

10

, we

conc

lude

that

sin

30

°=

10

20

=1 2

.

Now

wai

t jus

t a

min

ute,

tha

t’s v

ery

inte

rest

ing!

Why

did

a d

iffer

ent

size

tria

ngle

giv

e us

th

e sa

me

resu

lt? T

his

is m

ost u

nexp

ecte

d!

Inde

ed, m

uch

of m

athe

mat

ics

was

disc

over

ed in

the

sam

e w

ay. I

t st

arts

with

a s

impl

e cu

riosit

y, a

nd th

en a

con

scio

us e

ffort

to p

oke

and

prod

wha

teve

r you

hav

e un

til it

giv

es

up s

ome

prof

ound

trut

h.ii T

hink

of m

ath

like

a pi

ñata

, whi

ch c

onta

ins

so m

uch

delic

ious

tr

uth.

But

the

onl

y w

ay t

o ge

t at

it is

to

swin

g at

it a

few

tim

es (

and

ofte

ntim

es t

his

swin

ging

is tr

uly

diffi

cult!

).

In th

is ca

se, t

he p

rofo

und

trut

h se

ems

to b

e th

at s

in3

0°

=1 2 n

o m

atte

r wha

t siz

e tr

iang

le

we

have

. Thu

s w

e sh

are

the

follo

win

g ve

ry im

port

ant t

heor

em.

The

depe

nden

cy o

f sin𝛼

Th

e ou

tput

of s

in𝛼

onl

y de

pend

s on

𝛼. T

he s

ize

of th

e tr

iang

le is

irre

leva

nt.

Ther

e is

, of c

ours

e, a

rea

son

that

sin𝛼

doe

s no

t dep

end

on th

e si

ze o

f the

tria

ngle

, and

w

e’ll

brie

fly e

xplo

re th

is in

the

Exer

cise

s.

This

prof

ound

trut

h w

ill a

llow

us

to c

onst

ruct

any

siz

e tr

iang

le w

e w

ish

whe

n w

e go

abo

ut

dete

rmin

ing

the

sine

of a

ny a

ngle

. Thi

s is

use

ful,

since

som

e ch

oice

s w

ill b

e ea

sier

tha

n ot

hers

. In

fact

, it i

s th

is v

ery

fact

that

will

mot

ivat

e ou

r wor

k in

the

next

uni

t. Le

t’s b

riefly

ex

plor

e th

is id

ea.

Exam

ple

1d

Eval

uate

sin4

5°.

Base

d on

wha

t w

e ju

st le

arne

d, w

e sh

ould

cre

ate

a 4

5°−4

5°−

90

° tr

iang

le, a

nd t

hen

crea

te a

ratio

usi

ng th

e le

ngth

s of

the

oppo

site

side

and

the

hypo

tenu

se. I

t wou

ld b

e fa

ir to

ask

wha

t tria

ngle

, exa

ctly

, you

sho

uld

crea

te. W

e st

art w

ith th

e ba

sics

in F

igur

e 30

.

ii A

nd th

is pr

oces

s of

exp

erim

enta

tion

and

verif

icat

ion

can

be a

s qu

ick

and

pain

less

as

we

just

witn

esse

d,

or it

cou

ld ta

ke s

ever

al y

ears

. Or c

entu

ries.

But i

t’s th

e jo

urne

y th

at c

ount

s, rig

ht?

§2

The

Sin

e, C

osin

e, a

nd T

ange

nt fu

nctio

ns –

97

We’

ve d

raw

n so

me

isosc

eles

righ

t tria

ngle

. Now

how

sho

uld

we

labe

l the

ang

les

and

the

side

leng

ths?

Cer

tain

ly, t

he tw

o no

n-rig

ht a

ngle

s m

ust b

e 4

5°

(sin

ce th

ey a

re b

oth

equa

l an

d m

ust e

qual

90

°), b

ut w

hat a

bout

the

side

leng

ths?

We

coul

d, fo

r exa

mpl

e, s

tart

with

th

e le

gs, a

nd la

bel e

ach,

say

, 10

. Onc

e w

e’ve

mad

e th

at d

ecisi

on, w

e m

ust c

oncl

ude

that

th

e hy

pote

nuse

is 1

0√2

.iii W

e dr

aw th

is in

Fig

ure

31.

Ther

e’s

noth

ing

wro

ng w

ith t

he p

revi

ous

pict

ure,

but

it

also

isn

’t th

e be

st. S

houl

d w

e ch

oose

a s

mal

l num

ber?

Or,

shou

ldn’

t we

choo

se s

ome

num

ber w

here

we

won

’t ha

ve a

ny

simpl

ifica

tion?

Beca

use

of th

is, w

hy d

on’t

we

let t

he le

gs b

e of

leng

th 1

.iv T

hen

we

have

Fig

ure

32.

iii S

ee th

e pr

evio

us s

ectio

n.

iv Y

ou m

ay n

ot s

ee th

at th

is is

the

“sim

ples

t” n

umbe

r. Th

at’s

fine!

Thi

s is

som

ethi

ng y

ou fi

nd o

ut th

roug

h ex

perim

enta

tion

and

expl

orat

ion.

Figu

re 3

0

Figu

re 3

1

Uni

t fou

r – 9

8

To c

alcu

late

sin4

5°,

then

, we

just

nee

d to

set

up

our r

atio

acc

ordi

ngly

.v We

have

sin4

5°

=1 √2

.

Wha

t if w

e ha

d ch

osen

the

orig

inal

tria

ngle

, whe

re th

e le

gs h

ad a

leng

th o

f 10

? Th

en w

e w

ould

hav

e ha

d

sin4

5°

=1

0

10√2

=1 √2

,

Whi

ch is

the

sam

e an

swer

, alth

ough

it w

as o

ne le

ss s

tep

to g

et to

the

ans

wer

. In

futu

re

sect

ions

, we’

ll be

pic

king

a v

ery

part

icul

ar tr

iang

le to

wor

k w

ith w

hen

we

wan

t to

eval

uate

so

me

Trig

func

tion.

We

are

just

bei

ng c

leve

r, an

d tr

ying

to

save

our

selv

es s

ome

time

– al

thou

gh th

is is

not m

anda

tory

, and

you

are

free

to p

ick

any

tria

ngle

you

like

.vi

A sin

gle

step

may

see

m in

sign

ifica

nt, b

ut s

ince

we’

ll be

goi

ng b

ack

to th

ese

two

spec

ial

tria

ngle

s ag

ain

and

agai

n,vi

i you

’ll w

ant

to s

trea

mlin

e th

e pr

oces

s as

muc

h as

pos

sible

. Re

gard

less

, we

now

hav

e an

othe

r va

lue

or r

atio

to

add

to o

ur li

st, w

hich

you

’ll w

ork

on

com

plet

ing

in th

e Ex

erci

ses.

One

mor

e no

te b

efor

e w

e co

ntin

ue o

n: M

ost t

extb

ooks

will

tell

you

that

sin4

5°

=√

2 2. A

re

they

wro

ng?

And

whe

re d

id t

hey

get

that

ans

wer

fro

m?

And

why

wou

ld t

hey

both

er

writ

ing

it lik

e th

at?

Goo

d qu

estio

ns!

Let u

s now

mov

e on

to th

e Co

sine

func

tion.

By

its v

ery

nam

e, w

e ca

n in

fer t

hat i

t is r

elat

ed

to th

e sin

e fu

nctio

n, a

nd –

as

we’

ll so

on s

ee –

this

is v

ery

muc

h th

e ca

se.

v W

ill it

mat

ter w

hich

45

° ang

le w

e ch

oose

? Se

e fo

r you

rsel

f! vi W

e w

ill re

itera

te th

is po

int i

n fu

ture

sec

tions

. vi

i And

aga

in!

Figu

re 3

2

§2

The

Sin

e, C

osin

e, a

nd T

ange

nt fu

nctio

ns –

99

The

Cosin

e fu

nctio

n

Th

e Co

sine

func

tion

acce

pts

as a

n in

put

an a

cute

ang

le o

f a

right

tria

ngle

, and

the

n re

turn

s th

e ra

tio o

f the

sid

e ad

jace

nt to

the

angl

e to

the

hypo

tenu

se. S

ymbo

lical

ly,

cos𝛼

=𝑏 𝑐

.

We

coul

d al

so s

tate

that

co

s𝛽

=𝑎 𝑐

.

The

Cosin

e fu

nctio

n is

ver

y sim

ilar

to t

he S

ine

func

tion.

It, t

oo, a

ccep

ts a

n an

gle

as a

n in

put.

But

this

time

it re

turn

s as

an

outp

ut t

he s

ide

leng

th a

djac

entvi

ii to

𝛼 o

ver

the

hypo

tenu

se. T

his

– ag

ain

– ha

s on

ly b

een

defin

ed fo

r acu

te a

ngle

s in

a ri

ght t

riang

le.

Exam

ple

2a

Writ

e th

e Co

sine

ratio

of 𝛼

giv

en F

igur

e 33

.

All w

e ne

ed to

do

is m

ake

a fra

ctio

n of

the

leng

th o

f the

adj

acen

t leg

ove

r the

hyp

oten

use.

Th

is is

sim

ple:

We

get

vi

ii Not

e th

at a

djac

ent m

eans

touc

hing

. And

whi

le, y

es, t

he h

ypot

enus

e is

touc

hing

𝛼, w

hat w

e re

ally

mea

n is

the

adja

cent

leg.

Figu

re 3

3

Uni

t fou

r – 1

00

cos𝛼

=4 5

.

It is

impo

rtan

t to

note

that

thes

e ra

tios

will

onl

y w

ork

with

righ

t tria

ngle

s. W

as th

e ab

ove

tria

ngle

a ri

ght t

riang

le?

It m

ight

look

that

way

, but

how

can

we

be su

re?ix

Up

to th

is po

int,

we’

ve a

lway

s se

en t

he b

ox a

t th

e rig

ht a

ngle

. Her

e w

e do

n’t

have

one

– s

o is

our

res

ult

inco

rrec

t?

This

is a

n ex

celle

nt q

uest

ion

and

one

you

mus

t be

awar

e of

. Pyt

hago

ras’

Theo

rem

tells

us

that

for a

ny ri

ght t

riang

le,

𝑎2

+𝑏

2=𝑐

2.

The

conv

erse

of

Pyth

agor

as’ T

heor

em –

tha

t is,

if 𝑎

2+𝑏

2=𝑐

2 t

hen

you

have

a r

ight

tr

iang

le –

is a

lso tr

ue. S

ince

32

+4

2=

52

,

we

can

conc

lude

we

have

a r

ight

tria

ngle

. Th

us t

he a

fore

men

tione

d co

nclu

sion

(that

co

s𝛼

=4 5) i

s co

rrec

t.

Exam

ple

2b

List

out

the

Sine

and

Cos

ine

ratio

s fo

r ang

les 𝛼

and

𝛽 in

Fig

ure

34.

Rem

embe

r to

first

ver

ify th

at y

ou h

ave

a rig

ht tr

iang

le. S

ince

52

+12

2=

13

2,

we

can

mov

e on

to th

e ne

xt s

tep.

ix F

or e

xam

ple,

if th

e un

nam

ed a

ngle

, cal

l it 𝛾

, is

equa

l to

89

.99

(whi

ch is

real

ly c

lose

to 9

0) t

hen

our T

rig

ratio

s ar

e in

accu

rate

. We

mus

tbe

posi

tive

we

have

an

actu

al ri

ght t

riang

le; i

t is

not g

ood

enou

gh to

al

mos

t be

a rig

ht tr

iang

le.

Figu

re 3

4

§2

The

Sin

e, C

osin

e, a

nd T

ange

nt fu

nctio

ns –

101

Let u

s de

al w

ith 𝛼

firs

t. O

ur ra

tios

for 𝛼

are

sin𝛼

=5 13

and

cos𝛼

=12

13

.

Then

, with

resp

ect t

o 𝛽

, we

have

sin𝛽

=12

13

and

cos𝛽

=5 13

.

But i

sn’t

ther

e so

met

hing

pec

ulia

r abo

ut o

ur re

sults

? Lo

ok a

gain

– is

ther

e an

ythi

ng th

at

you

can

see?

Why

is it

that

sin𝛼

=5 13

=co

s𝛽

?

Or t

hat

sin𝛽

=12

13

=co

s𝛼

?

This

is a

not

e-w

orth

y fin

d! W

hy s

houl

d tw

o di

ffere

nt f

unct

ions

(w

ith d

iffer

ent

inpu

ts)

prod

uce

the

sam

e ou

tput

?

Befo

re w

e fo

rmal

ize

our n

ext t

heor

em, l

et u

s con

sider

the

rela

tions

hip

that

exi

sts b

etw

een

the

two

non-

right

ang

les

in a

righ

t tria

ngle

.

Firs

t, le

t us

(in F

igur

e 35

) dra

w s

ome

right

tria

ngle

.

Fi

gure

35

Uni

t fou

r – 1

02

Let

us la

bel t

he t

wo

non-

right

ang

les

as 𝛼

and

𝛽. I

s th

ere

som

e re

latio

nshi

p be

twee

n th

em?

We

now

nee

d so

me

sort

of s

tart

ing

poin

t –

so le

t’s t

hink

bac

k to

oth

er r

elat

ions

hips

in

tria

ngle

s. Th

e bi

gges

t on

e, p

erha

ps, i

s th

e th

eore

m th

at th

e su

m o

f all

thre

e an

gles

in a

tr

iang

le a

re 1

80

°. So

let’s

beg

in w

ith th

is, p

lay

with

it a

bit,

and

then

see

if w

e ca

n’t u

ncov

er

som

ethi

ng o

f not

e.x

Usin

g th

e pr

evio

us re

latio

nshi

p, w

e w

ould

then

hav

e

𝛼+𝛽

+𝛾

=1

80

.

But w

e kn

ow 𝛾

=9

0°,

since

it is

a ri

ght a

ngle

(and

by

defin

ition

, all

right

ang

les e

qual

90

°).

Subs

titut

ing

this

in, w

e se

e th

at

𝛼+𝛽

=9

0°.

This

yiel

ds u

s on

e im

port

ant

rela

tions

hip:

𝛼

and

𝛽 a

re c

ompl

emen

tary

. Th

is se

ems

beni

gn…

But

let u

s co

nsid

er th

e w

ord

Cosin

e fo

r a m

omen

t. It

is ac

tual

ly a

shor

tene

d fo

rm

of th

e te

rm “

Com

plem

ent

of t

he s

ine.

” W

ait a

mom

ent…

Is it

a c

oinc

iden

ce th

at w

e ju

st

show

ed th

at 𝛼

and

𝛽 a

re c

ompl

emen

tary

and

the

Cosin

e fu

nctio

n ha

s in

its v

ery

nam

e th

e w

ord

“com

plem

ent”

? M

ost

cert

ainl

y no

t!xi W

e’ve

disc

over

ed s

ome

prof

ound

tru

th!

We

now

pro

vide

a fo

rmal

pro

of.

Proo

f.

We

wan

t to

sh

ow

that

th

e tw

o no

n-rig

ht

angl

es

of

a rig

ht

tria

ngle

m

ust

be

com

plem

enta

ry, t

hat i

s, th

at th

eir s

um m

ust b

e 9

0°.

Cons

truc

t rig

ht Δ𝐴𝐵𝐶

with

righ

t ang

le 𝐵

.

x O

f cou

rse,

our

exp

lora

tion

may

turn

out

frui

tless

. Tha

t’s O

K! B

elie

ve it

or n

ot, f

indi

ng o

ut w

hat d

oesn

’t w

ork

can

also

ver

y he

lpfu

l. Ad

ditio

nally

, don

’t ge

t dis

cour

aged

if y

ou’re

not

cor

rect

the

first

tim

e –

very

fe

w m

athe

mat

icia

ns a

re!

xi W

hich

doe

sn’t

mea

n “N

ever

”. It

just

mea

ns u

nlik

ely.

Som

e m

ay s

coff

at th

e si

mpl

e lo

gic

we

used

her

e,

but k

eep

in m

ind

we’

re s

how

ing

a pr

oces

s of

com

ing

to a

n in

form

al c

oncl

usio

n. Y

ou m

ay w

ish th

e ex

plan

atio

n w

as m

ore

form

al, a

nd if

that

’s th

e ca

se, I

com

men

d yo

u an

d en

cour

age

you

to w

rite

your

ow

n te

xtbo

ok.

§2

The

Sin

e, C

osin

e, a

nd T

ange

nt fu

nctio

ns –

103

Beca

use

the

sum

of a

ll th

ree

angl

es in

a tr

iang

le is

18

0°,

we

know

that

𝐴+𝐵

+𝐶

=1

80

°. Bu

t 𝐵=

90

°, sin

ce (b

y de

finiti

on) a

ll rig

ht a

ngle

s ar

e eq

ual t

o 9

0°.

Then

, by

subs

titut

ion,

w

e ha

ve th

at 𝐴

+9

0°

+𝐶

=1

80

°. U

sing

the

subt

ract

ion

prop

erty

of e

qual

ity re

veal

s th

at

𝐴+𝐶

=9

0°,

whi

ch is

pre

cise

ly w

hat t

he w

ord

“com

plem

enta

ry” m

eans

.

We

have

thus

sho

wn

wha

t we

wan

ted

to s

how

.

With

this

in m

ind,

let u

s fo

rmal

ize

the

prev

ious

.

The

rela

tions

hip

betw

een

the

Sine

and

Cos

ine

func

tions

. Th

e tw

o no

n-rig

ht a

ngle

s in

a ri

ght t

riang

le a

re a

lway

s co

mpl

emen

tary

. Ex

ampl

e 2c

Eval

uate

co

s3

0°.

We

go th

roug

h th

e sa

me

proc

edur

e fo

und

in E

xam

ples

1c

and

1d. W

e fir

st d

raw

a ri

ght

tria

ngle

xii (

Figu

re 3

6).

Base

d on

the

tria

ngle

that

we

chos

e, w

e ha

ve th

at

cos𝛼

=ad

jace

nt

1.

xi

iRe

mem

ber t

hat w

e ar

e fr

ee to

dra

w a

ny ri

ght t

riang

le w

e w

ant.

Wel

l, as

long

as

is it

has

a 3

0° a

ngle

, an

yway

.

Figu

re 3

6

We

chos

e to

mak

e ou

r hyp

oten

use

1 h

ere.

The

re a

re a

few

oth

er n

ice

choi

ces,

as w

ell.

Uni

t fou

r – 1

04

But w

hat i

s th

e le

ngth

of t

he le

g ad

jace

nt to

𝛼?

We

can

figur

e th

is ou

t bec

ause

this

is a

spec

ial r

ight

tria

ngle

, viz

., it

is a

30

°−

60

°−

90

° tr

iang

le. S

ince

our

hyp

oten

use

is 1

, the

sh

ort

leg

(not

mar

ked)

mus

t mea

sure

1 2. T

hen,

to

find

the

long

leg,

we

mul

tiply

1 2 b

y √

3.

Thus

the

long

leg,

𝑎, m

ust b

e √3 2. S

ince

𝑎=

√3 2, w

e co

nclu

de th

at

cos𝛼

=

√3 2 1

=√

3 2.

We

have

one

mor

e Tr

igon

omet

ric f

unct

ion

to u

ncov

er. W

e’ll

defin

e it

here

,but

in t

he

exer

cise

s yo

u w

ill d

eriv

exiii i

t you

rsel

f.

The

Tang

ent f

unct

ion

Th

e Ta

ngen

t fun

ctio

n ac

cept

s as

an

inpu

t an

acut

e an

gle,

and

then

retu

rns

the

ratio

of

the

side

oppo

site

to th

e an

gle

to th

e si

de a

djac

ent t

o th

e an

gle.

Sym

bolic

ally

, ta

n𝛼

=𝑎 𝑏

.

We

coul

d al

so s

tate

that

tan𝛽

=𝑏 𝑎

.

The

Tang

ent f

unct

ion

appe

ars

a bi

t diff

eren

t fro

m th

e Si

ne a

nd C

osin

e fu

nctio

n, a

nd, t

o be

sur

e, th

ere

are

som

e ke

y di

ffere

nces

. But

you

’ll a

lso d

iscov

er in

the

exer

cise

s th

at it

is

clos

ely

rela

ted

to t

he S

ine

and

Cosin

e fu

nctio

ns! A

dditi

onal

ly –

as

you’

ll so

on s

ee –

we

wor

kw

ith it

in th

e sa

me

man

ner.

Exam

ple

3a

Writ

e th

e Si

ne, C

osin

e, a

nd T

ange

nt ra

tios

usin

g 𝛼

of t

he tr

iang

le s

how

n in

Fig

ure

37.

xi

ii Wha

t thi

s m

eans

is th

at y

ou’ll

take

som

ethi

ng y

ou a

lread

y kn

ow, w

ork

with

it, a

nd c

ome

out w

ith th

e fu

nctio

n w

e’re

abo

ut to

def

ine!

Obv

ious

ly th

is is

the

bett

er w

ay to

do

it, b

ut w

e do

n’t w

ant t

o sp

oil y

our

fun.

§2

The

Sin

e, C

osin

e, a

nd T

ange

nt fu

nctio

ns –

105

All w

e ne

ed to

do

is s

ubst

itute

. The

onl

y pr

oble

m w

e ha

ve is

that

we

don’

t hav

e a

leng

th

for

the

hypo

tenu

se. N

o m

atte

r –

this

is e

asily

foun

d. W

e us

e Py

thag

oras

’ The

orem

and

fin

d th

at th

e hy

pote

nuse

mus

t hav

e a

leng

th o

f 8.54

. With

this

in m

ind,

we

have

sin𝛼

=8

8.54

,

cos𝛼

=3 8.4

,

and

tan𝛼

=8 3

.

Exam

ple

3b

Eval

uate

tan4

5°.

We

follo

w t

he s

ame

proc

edur

e as

Exa

mpl

es 2

c, 1

c, a

nd 1

d. W

e cr

eate

any

siz

e rig

ht

tria

ngle

with

a 4

5° a

ngle

, as

in F

igur

e 38

.

Now

all

we

need

to d

o is

writ

e ou

r rat

io. W

e ha

ve

tan4

5°

=1 1

=1

.

Thus

we

conc

lude

that

tan4

5° i

s 1

.

Figu

re 3

7

Figu

re 3

8

Reca

ll th

at a

ny 4

5°−4

5°−

90

° tria

ngle

will

also

be

isosc

eles

. Bot

h le

gs, t

here

fore

, are

also

con

grue

nt.

Uni

t fou

r – 1

06

Usu

ally

, eva

luat

ing

a Tr

ig f

unct

ion

give

n an

y in

put

is v

ery

diffi

cult

to d

o. O

f co

urse

, in

toda

y’s

wor

ld, w

e ha

ve r

eady

acc

ess

to c

alcu

lato

rs, w

hich

are

ver

y go

od a

t th

e te

diou

s pr

oces

s of

app

roxi

mat

ing,

so

it isn

’t to

o di

fficu

lt to

eva

luat

e si

n2

°, f

or e

xam

ple.

You

’ll

need

to h

ave

a ca

lcul

ator

on

hand

.

Exam

ple

4a

Det

erm

ine

the

leng

th o

f 𝑎 g

iven

Fig

ure

39.

As w

e di

d ex

tens

ivel

y in

Geo

met

ry,

we

shou

ld f

irst

esta

blish

som

e re

latio

nshi

p. F

or

exam

ple,

we

know

(fro

m P

ytha

gora

s) t

hat 𝑎

2+𝑏

2=𝑐

2. I

n th

is ca

se, h

owev

er, w

e do

n’t

know

𝑎 o

r 𝑏, a

nd s

o Py

thag

oras

’ The

orem

yie

lds

us n

o he

lp. S

ince

we

have

bee

n gi

ven

an

angl

e m

easu

re a

nd o

ne si

de le

ngth

, it s

tand

s to

reas

on th

at w

e sh

ould

use

a T

rig fu

nctio

n.

We

have

thre

e di

ffere

nt c

hoic

es (S

ine,

Cos

ine,

and

Tan

gent

), so

whi

ch o

ne sh

ould

we

use?

In

thi

s ex

ampl

e, w

e w

ill tr

y al

l of t

hem

and

sho

w y

ou t

hat

ther

e is

rea

lly o

nly

one

good

ch

oice

.xiv

Firs

t, le

t us

try

the

Sin

e fu

nctio

n. G

iven

tha

t si

n𝛼

=opposite

hypotenuse

, we

have

(af

ter

subs

titut

ion)

sin2

0°

=𝑎 45

.

If w

e tr

y th

e Co

sine

func

tion,

we

have

cos2

0°

=ad

jace

nt

hyp

ote

nu

se=

? 45

,

whe

re th

e qu

estio

n m

ark

repr

esen

ts th

e un

know

n le

ngth

of t

he a

djac

ent.

If w

e tr

y th

e Ta

ngen

t fun

ctio

n, w

e w

ould

hav

e

tan2

0°

=o

pp

osi

te

adja

cen

t=𝑎 ?

,

whe

re, a

gain

, the

que

stio

n m

ark

repr

esen

ts th

e sa

me

unkn

own

leng

th o

f the

adj

acen

t.

xi

v Thi

s w

on’t

alw

ays

be th

e ca

se. S

omet

imes

ther

e w

ill b

e tw

o or

thre

e go

od c

hoic

es.

Figu

re 3

9

§2

The

Sin

e, C

osin

e, a

nd T

ange

nt fu

nctio

ns –

107

Base

d on

wha

t w

e ju

st s

aw, t

he S

ine

is t

he fu

nctio

n w

e w

ill c

hoos

e. T

his

is b

ecau

se w

e ha

ve a

n eq

uatio

n,

sin2

0°

=𝑎 45

,

whi

ch c

an b

e so

lved

for t

he m

issin

g va

riabl

e 𝑎

. If w

e us

ed th

e Co

sine

func

tion,

we

coul

d fin

d th

e le

ngth

of t

he a

djac

ent,

but t

hat w

ould

not

get

us

the

leng

th o

f 𝑎.xv

The

Tan

gent

fu

nctio

n is

even

wor

se; w

e w

ould

hav

e tw

o va

riabl

es in

our

equ

atio

n, w

hich

wou

ld p

ut u

s in

an

unte

nabl

e po

sitio

n.

To s

olve

this

equa

tion,

we

shou

ld e

valu

ate

the

left

side,

and

then

get

the

varia

ble

all b

y its

elf.

Usin

g a

calc

ulat

or, w

e fin

d th

at s

in2

0°≈

0.342

. Hen

ce

0.342

=𝑎 45

15

.39≈𝑎

.

The

key

to th

ese

prob

lem

s is

to id

entif

y th

e co

rrec

t rel

atio

nshi

p. T

hen

it’s

just

a m

atte

r of

subs

titut

ing

and

solv

ing,

bot

h of

whi

ch a

re b

asic

Alg

ebra

ski

lls.

Exam

ple

4b

Det

erm

ine

the

leng

th o

f 𝑏 g

iven

Fig

ure

40.

We

face

a s

imila

r pro

blem

. Her

e th

e be

st re

latio

nshi

p to

use

is T

ange

nt, s

ince

we

can

then

fo

rm th

e eq

uatio

n

tan

50

°=

7.8 𝑏

.

Then

we

just

sol

ve th

e eq

uatio

n w

henc

e

xv

That

bei

ng s

aid,

we

coul

d fin

d th

e le

ngth

of 𝑎

afte

r we

find

the

leng

th o

f the

adj

acen

t. It’

s no

t an

idea

l pr

oces

s th

ough

, as

we

have

to g

o th

roug

h th

e en

tire

proc

edur

e ag

ain,

or,

alte

rnat

ivel

y, u

se P

ytha

gora

s’ Th

eore

m. E

ither

way

, it’s

muc

h m

ore

effic

ient

to u

se th

e Si

ne fu

nctio

n.

Figu

re 4

0

Uni

t fou

r – 1

08

𝑏≈

6.54

.

Exam

ple

4c

Com

plet

e th

e rig

ht tr

iang

le s

how

n in

Fig

ure

41.

Reca

ll th

at w

hen

we

are

aske

d to

“com

plet

e” a

righ

t tria

ngle

, it j

ust m

eans

that

we

shou

ld

find

out t

he le

ngth

of e

very

sid

e an

d th

e m

easu

re o

f eve

ry a

ngle

.xvi

We

have

som

e fre

edom

in

how

we

appr

oach

thi

s pr

oble

m,

but

ther

e ar

e al

so s

ome

limita

tions

.xvii L

et u

s fir

st fi

nd 𝑎

. To

do th

is, w

e m

ust a

sk th

at e

ver-

popu

lar q

uest

ion:

Wha

t is

the

rela

tions

hip?

In th

is ca

se, w

e co

uldxv

iii u

se th

e Si

ne fu

nctio

n to

get

sin

58

°=

𝑎 12

0.8

5≈

𝑎 12

𝑎≈

10

.18

.

Now

that

we

have

𝑎, w

e ca

n ea

sily

find 𝑏

by

usin

g Py

thag

oras

’ The

orem

. We

can

also

find

𝛽

eas

ily, s

ince

the

sum

of t

he th

ree

angl

es m

ust b

e 1

80

°. H

ence

𝑏≈

6.3

5,𝛽

=32

°.

Befo

re y

ou b

egin

pra

ctic

ing

thes

e co

ncep

ts, t

here

are

thr

ee o

ther

Trig

fun

ctio

ns w

hich

ex

ist. T

hese

are

rare

ly u

sed

by th

emse

lves

, but

they

will

be

of s

ome

impo

rtan

ce la

ter.

We

men

tion

them

her

e fo

r the

sak

e of

com

plet

ion.

xv

i Tha

t mak

es a

tota

l of s

ix th

ings

we’

re lo

okin

g fo

r, al

thou

gh w

e m

ust b

e gi

ven

at le

ast t

hree

of t

hem

to

dete

rmin

e th

e ot

her t

hree

. xv

ii For

exa

mpl

e, c

ould

we

find

any

of th

e si

des

usin

g Py

thag

oras

’ The

orem

? xv

iii W

e sa

y “c

ould

” bec

ause

we

coul

d al

so fi

nd th

e le

ngth

of 𝑏

by

usin

g th

e Co

sine

func

tion.

Cou

ld w

e us

e th

e Ta

ngen

t fun

ctio

n he

re?

Why

or w

hy n

ot?

Figu

re 4

1

§2

The

Sin

e, C

osin

e, a

nd T

ange

nt fu

nctio

ns –

109

The

reci

proc

al T

rigon

omet

ric fu

nctio

ns

Th

e Co

seca

nt fu

nctio

n:

csc𝛼

=𝑐 𝑎

The

Seca

nt fu

nctio

n:

sec𝛼

=𝑐 𝑏

The

Cota

ngen

t fun

ctio

n:

cot𝛼

=𝑏 𝑎

The

nam

e “r

ecip

roca

l Trig

func

tions

” is

very

apr

opos

. See

if y

ou c

an fi

nd o

ut w

hy.

§𝟐 E

xerc

ises

1.)

Writ

e th

e ra

tios

of th

e Si

ne, C

osin

e, a

nd T

ange

nt fu

nctio

n (fo

r bot

h 𝛼

and

𝛽) g

iven

th

e fo

llow

ing

tria

ngle

s.

(A)

(B)

(C)

(D)

2.)

Our

nex

t goa

l is

to c

reat

e a

tabl

e fo

r the

Sin

e, C

osin

e, a

nd T

ange

nt fu

nctio

ns. F

irst,

how

ever

, we’

ll on

ly fo

cus

on t

hose

rea

lly n

ice

angl

es w

hich

pro

duce

spe

cial

rig

ht

tria

ngle

s. (A

) W

hat

are

the

“rea

lly n

ice

angl

es”

of w

hich

we

spea

k? (

Hin

t: T

hink

of

the

angl

es s

een

in a

ll of

our

spe

cial

righ

t tria

ngle

s.)

Uni

t fou

r – 1

10

(B)

Dra

w a

ny tw

o tr

iang

les

with

the

info

rmat

ion

from

(A).

Mak

e su

re y

ou la

bel

the

sides

and

ang

les.

(C)

Now

cre

ate

a ta

ble

of v

alue

s fo

r th

e Si

ne, C

osin

e, a

nd T

ange

nt f

unct

ions

. U

se (a

nd e

xpan

d) th

e in

com

plet

e ta

ble

belo

w a

s a

mod

el.

𝜶

𝐬𝐢𝐧𝜶

3

0°

1 2

45

° √2 2

3.)

We

have

not

yet

def

ined

sin

0°

and

cos

0°,

so l

et’s

do t

hat

now

. Co

nsid

er t

he

tria

ngle

bel

ow, w

here

𝛼 is

a v

ery

smal

l ang

le.

Not

e th

at, i

n th

e ab

ove

case

, sin𝛼

will

be

rela

tivel

y sm

all (

whi

ch y

ou s

houl

d ve

rify)

.

(A)

Wha

t will

hap

pen

to th

e le

ngth

of t

he s

ide

oppo

site

of 𝛼

as

the

angl

e of

𝛼

decr

ease

s?

(B)

Fina

lly, i

mag

ine

that

𝛼=

0. W

hat

will

the

leng

th o

f the

sid

e op

posi

te t

o 𝛼

be

? (C

) G

iven

you

r ans

wer

for (

B), w

hat m

ust s

in0

° be

equa

l to?

(D

) N

ow l

et’s

cons

ider

the

Cos

ine

func

tion.

Alre

ady,

the

len

gth

of t

he s

ide

adja

cent

to 𝛼

and

the

leng

th o

f the

hyp

oten

use

are

very

clo

se. W

hat i

s co

s𝛼

gi

ven

the

pict

ure?

(E

) As

𝛼 g

ets c

lose

to z

ero,

wha

t will

hap

pen

to th

e le

ngth

of t

he h

ypot

enus

e in

re

latio

n to

the

leng

th o

f the

adj

acen

t sid

e? (H

int:

Try

dra

win

g sm

alle

r and

sm

alle

r ang

les

for 𝛼

) (F

) G

iven

you

r ans

wer

for (

E), w

hat m

ust c

os

0° b

e eq

ual t

o?

(G)

Verif

y yo

ur re

sults

for (

C) a

nd (F

) with

a c

alcu

lato

r. 4.

) N

ow le

t’s d

efin

e th

e re

sults

of s

in9

0° a

nd c

os

90

°. (A

) W

e sh

ould

try

to

use

a sim

ilar

appr

oach

as

the

prev

ious

pro

blem

. Dra

w a

rig

ht tr

iang

le w

here

𝛼 is

clo

se to

90

°, bu

t not

qui

te th

at la

rge.

(B

) Ba

sed

on y

our p

ictu

re, w

hat w

ill s

in9

0° a

nd c

os

90

° be?

(C

) Ve

rify

your

resu

lts w

ith a

cal

cula

tor.

5.)

Crea

te a

tabl

e of

val

ues

for a

ll th

ree

Trig

func

tions

from

0° t

o 9

0°.

Incr

emen

t eac

h ro

w b

y 5

° (s

o yo

ur f

irst

num

ber

is 0

°, th

en 5

°, th

en1

0°…

). Yo

u sh

ould

use

a

calc

ulat

or fo

r thi

s pr

oble

m.

§2

The

Sin

e, C

osin

e, a

nd T

ange

nt fu

nctio

ns –

111

6.)

Com

plet

e th

e fo

llow

ing

right

tria

ngle

s.

(A)

(B)

(C)

(D)

(E)

(F)

7.

) Th

e pr

evio

us c

once

pts

can

also

be

appl

ied

to q

uasi-

real

sce

nario

s. (A

) Be

nny

lean

s a

ladd

er a

gain

st a

24

foot

wal

l, so

tha

t th

e to

p of

the

ladd

er

touc

hes

the

top

of t

he w

all p

erfe

ctly

. The

ang

le t

hat

is cr

eate

d w

ith t

he

grou

nd a

nd th

e la

dder

is 7

0°.

Det

erm

ine

the

leng

th o

f the

ladd

er.

(B)

Build

ing

A is

sho

rter

than

Bui

ldin

g B.

To

dete

rmin

e ho

w m

uch

shor

ter,

the

may

or la

ys a

tape

mea

sure

from

Bui

ldin

g A

to B

uild

ing

B an

d se

es th

at th

e sla

nt h

eigh

t is 4

5 m

. The

n he

also

figu

res

out

that

the

ang

le b

etw

een

the

tape

mea

sure

and

Bui

ldin

g B

is 62

°. H

ow m

uch

shor

ter

is Bu

ildin

g A

than

Bu

ildin

g B?

(C

) A

wat

er sl

ide

is h

ouse

d in

side

a to

wer

that

is 5

0 ft

tall.

The

wat

er sl

ide

trav

els

in a

str

aigh

t lin

e do

wn

tow

ard

the

grou

nd, w

here

it h

its th

e gr

ound

at a

24

° an

gle.

How

muc

h sp

ace

is n

eede

d on

the

grou

nd fo

r thi

s sli

de?

(D)

A tr

ee’s

shad

ow is

10

ft lo

ng. T

he a

ngle

tha

t th

e su

n cr

eate

s w

ith t

he fl

at

grou

nd is

80

°. D

eter

min

e th

e he

ight

of t

he tr

ee.

8.)

Crea

te y

our o

wn

quas

i-rea

l situ

atio

n, s

imila

r to

one

of th

e pr

evio

us p

robl

ems,

that

re

quire

s Tr

igon

omet

ry to

sol

ve.

9.)

Look

at t

he ta

ble

of v

alue

s yo

u cr

eate

d fo

r the

Sin

e fu

nctio

n.

(A)

Wha

t is

the

max

imum

val

ue o

f sin𝛼

?

Uni

t fou

r – 1

12

(B)

Wha

t is

the

min

imum

val

ue o

f sin𝛼

? (C

) Ar

e th

ose

max

imum

and

min

imum

val

ues

true

for

any

val

ue o

f 𝛼

? Tr

y si

n5

00

° or s

in( −

10

0°)

to v

erify

. (D

) D

escr

ibe

any

patt

erns

you

see

with

the

Sine

func

tion.

10

.) Lo

ok a

t the

tabl

e of

val

ues

you

crea

ted

for t

he C

osin

e fu

nctio

n.

(A)

Wha

t is

the

max

imum

val

ue o

f co

s𝛼

? (B

) W

hat i

s th

e m

inim

um v

alue

of c

os𝛼

? (C

) Ar

e th

ose

max

imum

and

min

imum

val

ues

true

for a

ny v

alue

of 𝛼

? (D

) D

escr

ibe

any

patt

erns

you

see

with

the

Cosin

e fu

nctio

n.

11.)

Look

at t

he ta

ble

of v

alue

s yo

u cr

eate

d fo

r the

Tan

gent

func

tion.

(A

) W

hat i

s th

e m

axim

um v

alue

of t

an𝛼

? (B

) W

hat i

s th

e m

inim

um v

alue

of t

an𝛼

? (C

) Ar

e th

ose

max

imum

and

min

imum

val

ues

true

for a

ny v

alue

of 𝛼

? (D

) D

escr

ibe

any

patt

erns

you

see

with

the

Tang

ent f

unct

ion.

12

.) Im

agin

e fo

r a m

omen

t tha

t you

mea

sure

a tr

iang

le in

yar

ds, a

nd y

ou fi

nd th

at th

e tw

o le

gs m

easu

re 3

and 4

yar

ds, w

hile

the

hypo

tenu

se m

easu

res

5 y

ards

. You

sen

d th

is in

form

atio

n to

you

r frie

nd w

ho li

ves

in a

Eur

opea

n co

untr

y th

at n

orm

ally

use

s m

eter

s in

stea

d.

(A)

Calc

ulat

e si

n𝛼

usin

g th

e in

itial

cor

rect

mea

sure

men

t tha

t use

d ya

rds.

(B)

Now

con

vert

yar

ds in

to m

eter

s (ro

und

to th

e ne

ares

t hun

dred

th).

Calc

ulat

e si

n𝛼

in m

eter

s. (C

) As

sum

e th

at w

hen

you

sent

thi

s tr

iang

le t

o yo

ur fr

iend

, you

did

not

labe

l yo

ur m

easu

rem

ents

. So

your

frie

nd a

ssum

es y

ou m

ean

3 m

eter

s, 4

met

ers,

and

so o

n. C

alcu

late

sin𝛼

with

this

inco

rrec

t inf

orm

atio

n.

(D)

Was

you

r an

swer

any

diff

eren

t in

(A)

, (B)

, or

(C)?

Wha

t do

es t

hat

tell

you

abou

t Sin

e an

d Co

sine

ratio

s (v

iz.,

do u

nits

mat

ter)?

13

.) O

ne p

opul

ar m

nem

onic

dev

ice

for

rem

embe

ring

the

diffe

rent

Trig

rat

ios

is S

OH

CA

H T

OA.

Exp

lain

wha

t thi

s m

eans

and

how

it h

elps

you

to re

mem

ber h

ow to

set

up

you

r rat

ios.

(If y

ou d

on’t

know

it, l

ook

it up

) 14

.) Le

t us

now

der

ive

the

Tang

ent f

unct

ion.

To

acco

mpl

ish th

is, w

e on

ly n

eed

the

Sine

an

d Co

sine

func

tion.

(A

) W

hen

deriv

ing

thin

gs, i

t is

ofte

n us

eful

to r

ewrit

e th

em in

thei

r mos

t bas

ic

term

s. Be

fore

we

can

do th

at, h

owev

er, l

et u

s dr

aw a

righ

t tria

ngle

. Cho

ose

any

non-

right

ang

le a

nd c

all i

t 𝛼

. The

n la

bel t

he s

ides

(acc

ordi

ng t

o yo

ur

choi

ce o

f 𝛼) a

s “o

ppos

ite”,

“adj

acen

t”, a

nd “h

ypot

enus

e”.

(B)

Usin

g yo

ur p

ictu

re, w

rite

out t

he S

ine

and

Cosin

e ra

tios.

(C)

Is th

ere

any

way

we

can

take

our

Sin

e an

d Co

sine

ratio

and

turn

them

into

ou

r Ta

ngen

t ra

tio?

Reca

ll th

at T

ange

nt is

opposite

adjacent. T

ry a

ddin

g, s

ubtr

actin

g,

mul

tiply

ing,

or d

ivid

ing

them

.

§2

The

Sin

e, C

osin

e, a

nd T

ange

nt fu

nctio

ns –

113

(D)

Why

is it

bet

ter t

o de

rive

the

tang

ent f

unct

ion

than

to d

efin

e it?

Put

ano

ther

w

ay, w

hat i

ssue

s w

ould

you

run

into

if y

ou ju

st d

efin

ed e

very

thin

g in

mat

h?

§3

Re

latio

nshi

ps b

etw

een

the

Trig

func

tions

In o

ur f

inal

sec

tion,

we’

ll lo

ok a

t ho

w t

he t

hree

mai

n Tr

ig f

unct

ions

are

rel

ated

to

one

anot

her.

We

alre

ady

know

a f

ew r

elat

ions

hips

, vi

z. th

at t

he C

osin

e fu

nctio

n is

the

com

plem

ent t

o th

e si

ne fu

nctio

n. S

o it

is fa

ir to

bel

ieve

that

ther

e ar

e ot

her r

elat

ions

hips

th

at e

xist

, and

in th

is se

ctio

n, w

e w

ill te

ase

them

out

. We

will

spe

nd o

ur ti

me

deriv

ing

and

form

aliz

ing

man

y of

the

basic

and

fund

amen

tal i

dent

itie

s in

this

sect

ion.

An

iden

tity

is a

di

ffere

nt w

ay to

writ

e an

equ

ival

ent s

tate

men

t. Fo

r exa

mpl

e, 2

+2

=4

is a

n id

entit

y; 2

+2

is

sim

ply

a di

ffere

nt n

ame

for 4

.

Befo

re w

e em

bark

on

lear

ning

som

e id

entit

ies,

let u

s fir

st lo

ok a

t a m

ore

basic

que

stio

n:

Doe

s kn

owin

g on

e Tr

ig r

atio

lead

us

to f

ind

any

of t

he o

ther

s? P

ut a

noth

er w

ay, i

f w

e kn

ow o

ne T

rig ra

tio, c

an w

e fin

d al

l of t

he o

ther

?

Exam

ple

1a

If si

n𝛼

=3 5, w

rite

the

othe

r six

Trig

ratio

s.

Let

us f

irst

draw

a p

ictu

re.

Beca

use

the

Sine

rat

io i

s th

e sid

e op

posi

te t

o 𝛼

to

the

hypo

tenu

se, i

t mak

es s

ense

to d

raw

Fig

ure

42.

This

draw

ing

mak

es o

ur li

fe m

uch

easie

r. N

ow w

hat w

e’ll

do is

det

erm

ine

the

othe

r rat

ios.

Star

ting

with

Cos

ine,

we

wou

ld h

ave co

s𝛼

=ad

jace

nt

5.

Figu

re 4

2

As b

efor

e, th

is isn

’t th

e on

ly tr

iang

le th

at y

ou c

ould

dra

w.

Uni

t fou

r – 1

14

But w

hat i

s the

leng

th o

f the

adj

acen

t sid

e? T

his c

an b

e ea

sily

figur

ed o

ut w

ith P

ytha

gora

s’ Th

eore

m, a

lthou

gh, i

n th

is ca

se, w

e re

cogn

ize

that

we

have

a P

ytha

gore

an T

riple

, and

thus

th

e ad

jace

nt s

ide

is 4

. With

this

in m

ind,

the

rest

of t

he T

rig ra

tios

are

elem

enta

ry. T

hey

are

cos𝛼

=4 5

,ta

n𝛼

=3 4

,cs

c𝛼

=5 3

,se

c𝛼

=5 4

,co

t𝛼

=4 3

.

Exam

ple

1b

If ta

n𝛽

=5 7, w

rite

the

othe

r six

Trig

ratio

s.

You

shou

ld, a

gain

, sta

rt w

ith a

pic

ture

.i We

draw

the

tria

ngle

sho

wn

in F

igur

e 43

.

Befo

re w

e w

rite

out a

ny o

f our

ratio

s, ho

w a

bout

we

find

the

leng

th o

f the

miss

ing

side,

w

hich

in

the

pres

ent

case

is

the

hypo

tenu

se.

Sinc

e w

e do

not

rec

ogni

ze t

his

as a

Py

thag

orea

n Tr

iple

, we

go a

head

and

use

Pyt

hago

ras’

Theo

rem

. We

find

that

52

+7

2=𝑐

2

𝑐=√

74

.ii

With

this

valu

e, w

e ca

n no

w w

rite

all s

ix ra

tios:

sin𝛼

=5

√74

,co

s𝛼

=7

√74

,ta

n𝛼

=5 7

,cs

c𝛼

=√

74

5,

sec𝛼

=√

74

7,

cot𝛼

=7 5

.

i S

ensi

ng a

pat

tern

yet

? ii Y

ou s

houl

d al

way

s le

ave

this

num

ber i

n ex

act f

orm

. Sim

plify

it if

you

can

.

Figu

re 4

3

§3

Rel

atio

nshi

ps b

etw

een

the

Trig

func

tions

– 1

15

Not

e th

at it

isn’

t ne

cess

ary

to r

atio

naliz

e th

e de

nom

inat

ors,

alth

ough

it m

ight

be

good

pr

actic

e fo

r yo

u to

do

so. A

lso n

ote

that

som

e te

xtbo

oks

and

stan

dard

ized

tes

ts w

ill

requ

ire y

ou to

do

this,

so

mak

e su

re y

ou k

now

how

to d

o th

is.

Exam

ple

1c

Giv

enth

at s

in𝛼

=𝑎

, fin

d al

l six

Trig

ratio

s.

This

see

ms

mor

e di

fficu

lt th

an t

he p

revi

ous

prob

lem

s. D

o no

t le

t fir

st a

ppea

ranc

es

intim

idat

e yo

u –

you

just

nee

d to

dra

w a

pic

ture

. The

onl

y iss

ue w

ith o

ur p

ictu

re is

tha

t w

e on

ly s

eem

to h

ave

the

leng

th o

f one

sid

e, 𝑎

. But

reca

ll th

at 𝑎

=𝑎 1

, so

we

can

draw

our

pict

ure

as s

een

in F

igur

e 44

.

So w

hat’s

the

len

gth

of t

head

jace

nt s

ide?

We

can

figur

e th

is o

ut u

sing

Pyt

hago

ras’

Theo

rem

, whi

ch, u

nsur

prisi

ngly

, is

exac

tly w

hat w

e di

d in

the

prev

ious

Exa

mpl

es. W

e le

t 𝑏

be th

e le

ngth

of t

he a

djac

ent s

ide,

and

set

up

our e

quat

ion

like

so:

𝑎2

+𝑏

2=

12

.

We

will

let y

ou fi

nish

this

Exam

ple

in th

e Ex

erci

ses.

We

have

see

n ho

w w

e ca

n re

late

one

Trig

rat

io t

o th

e ot

hers

. We

will

nex

t re

late

the

fu

nctio

ns th

emse

lves

.

Ago

od s

tart

ing

poin

t is

to b

egin

with

wha

t we

know

, and

bui

ld fr

om th

ere.

We

know

that

(1)

cos𝛼

=si

n𝛽

iff 𝛼

+𝛽

=9

0°.

This

is a

fine

rela

tions

hip,

but

let’s

mak

e th

is m

ore

usef

ul. O

ne is

sue

is th

at th

ere

are

two

varia

bles

, so

let’s

try

to s

impl

ify th

at s

tate

men

t.

Figu

re 4

4

Uni

t fou

r – 1

16

Exam

ple

2a

Rew

rite

(1) i

n te

rms

of 𝛼

.iii

We

are

give

n th

at c

os𝛼

=si

n𝛽

iff 𝛼

+𝛽

=9

0°.

We

can

thus

mak

e a

subs

titut

ion.

Sin

ce

𝛼+𝛽

=9

0°

then

we

know

that

𝛽=

90

°−𝛼

and

henc

e

cos𝛼

=si

n( 9

0°−𝛼) .

Exam

ple

2b

Giv

enco

s2

0°,

wha

t is

the

equi

vale

nt c

ofun

ctio

n?

To a

nsw

er t

his,

we

need

onl

y us

e th

e pr

evio

us r

esul

t. In

the

pre

sent

cas

e, 𝛼

=2

0°

and

henc

e

cos2

0°

=si

n( 9

0°−2

0°)

=si

n7

0°.

The

answ

er w

e’re

loo

king

for

is

sin

70

°. A

calc

ulat

or c

an q

uick

ly v

erify

tha

t in

deed

, co

s2

0°

=si

n7

0°.

Do

reco

gniz

e th

at th

e co

func

tion

of C

osin

e is

Sin

e, a

nd v

ice

vers

a. W

hich

Trig

func

tion

is

the

cofu

nctio

n of

Tan

gent

? O

r how

abo

ut C

osec

ant?

Is th

ere

an e

asy

way

to te

ll?

Exam

ple

2c

Writ

e al

l of t

he c

ofun

ctio

ns id

enti

ties

.

We

will

not

writ

e al

l of t

hem

, but

inst

ead,

will

writ

e on

e of

them

and

leav

e th

e re

st to

the

read

er.

sin𝛼

=co

s(9

0°−𝛼) .

Oth

er c

ofun

ctio

n id

entit

ies

exist

, and

you

will

hav

e to

writ

e th

em o

ut in

you

r exe

rcis

es.

The

cofu

nctio

n id

entit

ies

can

be u

sefu

l, bu

t m

ostly

the

y ju

st h

ighl

ight

the

rel

atio

nshi

p be

twee

n th

e Si

ne a

nd C

osin

e fu

nctio

n (a

nd th

e ot

her c

o-Tr

ig fu

nctio

ns).

iii In

oth

er w

ords

, the

re s

houl

d on

ly b

e on

e va

riabl

e, 𝛼

.

§3

Rel

atio

nshi

ps b

etw

een

the

Trig

func

tions

– 1

17

Let u

s no

w e

xplo

re w

hat h

appe

ns w

hen

we

squa

re th

e Si

ne o

r Co

sine

func

tions

. Firs

t of

all,

if w

e w

rite

sin𝛼

2 th

ere

coul

d be

a b

it of

con

fusi

on.iv

So,

if w

e m

ean ( s

in𝛼)(

sin𝛼) ,

we

will

writ

e

sin

2𝛼

.

If w

e m

ean

sin( 𝛼

2) ,

whe

re t

he a

ngle

𝛼 is

squ

ared

but

not

the

func

tion,

we’

ll w

rite

it as

si

n( 𝑎

2) .

Exam

ple

3

Eval

uate

sin

2𝛼

giv

en F

igur

e 45

.

Reca

ll th

at s

in2𝛼

=( s

in𝛼)(

sin𝛼) ,

and

sin

ce s

in𝛼

=3 5, w

e ha

ve, b

y su

bstit

utio

n,

(3 5)(3 5)

9 25

.

You’

ll w

ork

to e

xplo

re s

quar

ed T

rig fu

nctio

ns in

the

exer

cise

s.

Now

let’s

see

wha

t hap

pens

whe

n w

e ad

d tw

o sq

uare

d fu

nctio

ns.

Let’s

try

sin

24

5°

+co

s24

5°.

Sinc

e si

n4

5°

=√

2 2, w

e ca

n co

nclu

de th

at

sin

24

5°

=2 4

=1 2

.

This

is a

lso

true

of c

os24

5°,

since

co

s4

5°

=√

2 2. T

hus

we

have

sin

24

5°

+co

s24

5°

=1 2

+1 2

=1

.

This

is q

uite

nic

e. W

ill th

is al

way

s be

true

?

iv A

re w

e sq

uarin

g th

e an

gle 𝛼

or s

quar

ing

the

func

tion

sin𝛼

?

Figu

re 4

5

Uni

t fou

r – 1

18

Exam

ple

4

Eval

uate

sin

26

0°

+co

s26

0°.

From

the

prev

ious

sec

tion,

we

have

mem

oriz

ed t

hat

sin

60

°=

√3 2.v T

hus

sin

26

0°

=3 4. W

e

also

hav

e m

emor

ized

co

s6

0°

=1 2, a

nd th

us c

os2

60

°=

1 4. W

e th

usly

con

clud

e th

at

sin

26

0°

+co

s26

0°

=3 4

+1 4

=1

.

But

that

’s st

rang

e… W

hy h

ave

we

gott

en t

he s

ame

answ

er?

And

why

is t

hat

answ

er s

o pl

easa

nt?

Hav

e w

e st

umbl

ed u

pon

som

e re

mar

kabl

e tr

uth?

Pyth

agor

ean

iden

tity

For a

ny a

ngle

𝛼,

sin

2𝛼

+co

s2𝛼

=1

. W

e’ll

revi

sit th

is id

entit

y in

gra

phic

al fo

rmat

, but

that

will

be

afte

r we

reve

al th

e U

nit C

ircle

. It

is th

ere

that

we

will

pro

ve th

is id

entit

y, a

s it i

s ver

y ea

sy to

do

whe

n w

e vi

ew it

gra

phic

ally

.

It w

ould

be

safe

to

ask

if th

is is

the

only

suc

h re

latio

nshi

p be

twee

n co

func

tions

. For

ex

ampl

e, is

tan

2𝛼

+co

t2𝛼

also

equ

al to

one

? Yo

u w

ill e

xplo

re th

at in

the

exer

cise

s.

In §2

, we

brie

fly i

ntro

duce

d th

e Re

cipr

ocal

Trig

fun

ctio

ns, s

uch

as S

ecan

t. Th

ese

are

defin

ed a

s th

e re

cipr

ocal

s of

thei

r res

pect

ive

Trig

func

tion.

In c

ase,

for e

xam

ple,

we

wan

t to

fin

d th

e ra

tio o

f th

e hy

pote

nuse

to

the

oppo

site

, we

wou

ld t

hen

use

the

Cose

cant

fu

nctio

n. A

nd s

ince

csc𝛼

=h

ypo

ten

use

op

po

site

whi

le

sin𝛼

=o

pp

osi

te

hyp

ote

nu

se,

we

reco

gniz

e th

at th

ese

are

reci

proc

als

(thus

the

nam

e).

Reca

ll th

at o

ne w

ay w

e ca

n w

rite

a re

cipr

ocal

is t

o pu

t it

unde

r on

e. L

et’s

put

that

into

sy

mbo

ls. If

𝑎 is

som

e nu

mbe

r, an

d w

e w

ant t

o fin

d its

reci

proc

al, w

e ca

n ju

st e

valu

ate

(or

v A

nd if

you

hav

en’t

mem

oriz

ed it

, tha

t’s o

k to

o. Y

ou w

ill th

en n

eed

to c

onst

ruct

a 3

0°−

60

°−

90

° tria

ngle

an

d w

rite

out t

he ra

tio, t

hen

sim

plify

wha

t you

hav

e.

§3

Rel

atio

nshi

ps b

etw

een

the

Trig

func

tions

– 1

19

leav

e, if

we

pref

er) 1 𝑎

. So

if w

e w

ant t

o fin

d th

e re

cipr

ocal

of t

he n

umbe

r 3 4, w

e ju

st n

eed

to

eval

uate

1 3 4

, whi

ch is

4 3, a

s yo

u sh

ould

ver

ify.

With

this

in m

ind

we

can

now

def

ine

the

Reci

proc

al id

entit

ies.

Reci

proc

al id

entit

ies

sin𝛼

=1

csc𝛼

,cs

c𝛼

=1

sin𝛼

Of c

ours

e, t

here

are

oth

er r

ecip

roca

l ide

ntiti

es, b

ut y

ou w

ill w

rite

the

rem

aind

er in

you

r ex

erci

ses.

Let u

s no

w p

ut th

ese

iden

titie

s to

use

.

Exam

ple

5a

Sim

plify

sin𝛼

csc𝛼

+cos𝛼

sec𝛼

.

The

key

to th

is p

robl

em is

to re

writ

e th

is e

xpre

ssio

n in

to s

omet

hing

eas

ier.

So w

hat w

e’ll

wan

t to

do

is id

entif

y so

me

aspe

ct o

f the

exp

ress

ion

that

can

be

rew

ritte

n. In

thi

s ca

se,

let’s

try

rew

ritin

g cs

c𝛼

and

see

wha

t we

get.

Sinc

e

sin𝛼

csc𝛼

=si

n𝛼∙

1

csc𝛼

and

1

csc𝛼

=si

n𝛼

, we

can

say

that

sin𝛼

csc𝛼

=si

n2𝛼

.

This

sam

e pr

oced

ure

will

reve

al th

at co

s𝛼

sec𝛼

=co

s2𝛼

.

So w

e ca

n re

writ

e ea

ch a

dden

d in

our

orig

inal

pro

blem

and

we

end

up w

ith

sin

2𝛼

+co

s2𝛼

,

whi

ch is

n’t t

oo b

ad. B

ut w

e ca

n re

writ

e th

at e

xpre

ssio

n as

som

ethi

ng e

ven

mor

e sim

ple!

Si

nce

that

is a

Pyt

hago

rean

iden

tity

and

is e

qual

to o

ne, o

ur fi

nal r

esul

t is

simpl

y

1.

How

did

we

know

to c

hang

e cs

c𝛼

and

sec𝛼

? N

othi

ng b

ut in

tuiti

on –

in o

ther

wor

ds, w

hen

we

wor

k w

ith t

hese

sor

ts o

f pr

oble

ms,

ther

e is

no

pres

crib

ed m

etho

d to

sim

plify

the

ex

pres

sion.

Exp

erie

nce

is a

big

hel

p, b

ut d

o no

t un

dere

stim

ate

plan

ning

and

pat

ienc

e,

Uni

t fou

r – 1

20

eith

er. A

nd o

f cou

rse,

you

mus

t be

wel

l-ver

sed

in th

e va

rious

iden

titie

s th

at w

e’ve

lear

ned

so fa

r. Yo

u’ll

find

that

if y

ou d

on’t

know

the

iden

titie

s ver

y w

ell t

hat s

impl

ifyin

gex

pres

sions

in

this

man

ner w

ill b

e ve

ry d

iffic

ult.

No

mat

ter h

ow g

ood

you

beco

me

at th

ese

iden

titie

s, ho

wev

er, b

e pr

epar

ed t

o sp

end

som

e tim

e w

ith t

hem

. Eve

n ve

ry g

ood

mat

hem

atic

ians

so

met

imes

stru

ggle

with

thes

e, so

don

’t fe

el in

com

pete

nt if

you

don

’t ge

t the

ans

wer

righ

t aw

ay.

One

rule

of t

hum

b w

hich

we’

ll re

itera

te to

you

: Yo

u w

ill m

ost l

ikel

y w

ant t

o co

nver

t any

Tr

ig fu

nctio

ns in

to S

ine

or C

osin

e, if

pos

sible

. We

have

man

y id

entit

ies

whi

ch w

ork

with

Si

ne a

nd C

osin

e, b

ut o

nly

a fe

w t

hat

wor

k w

ith t

he r

ecip

roca

l Trig

fun

ctio

ns. T

his

isn’t

alw

ays

the

case

, but

it’s

usua

lly th

e be

st p

lace

to s

tart

.

Exam

ple

5b

Sim

plify

sin𝛼

cos𝛼.

You

shou

ld h

ave

alre

ady

foun

d th

is ou

t in

the

Exer

cise

s in

the

prev

ious

sec

tion,

but

this

one

is so

impo

rtan

t tha

t goi

ng o

ver i

t a s

econ

d tim

e w

ill b

e he

lpfu

l to

you.

Not

e th

at th

e pr

oced

ure

we

use

here

will

ver

y ra

rely

be

used

by

you

in y

our

Exer

cise

s. Bu

t ag

ain,

the

re

sult

is s

o im

port

ant w

e fe

el it

nec

essa

ry to

incl

ude.

One

way

we

can

rew

rite

sin𝛼

and

co

s𝛼

is in

ter

ms

of t

heir

ratio

s. Bu

t ra

tios

requ

ire a

tr

iang

le, r

ight

? An

d w

e do

n’t

have

one

, so

wha

t sh

all w

e do

? W

ell,

how

abo

ut w

e m

ake

one?

Con

side

r Fig

ure

46, w

hich

will

allo

w u

s to

find

the

ratio

s of

our

two

Trig

func

tions

.

With

this

in m

ind,

we

can

now

sub

stitu

te, s

ince

sin𝛼

=𝑎 𝑐 a

nd c

os𝛼

=𝑏 𝑐, w

e ha

ve

sin𝛼

cos𝛼

=

𝑎 𝑐 𝑏 𝑐

.

Figu

re 4

6

This

isn’t

the

only

way

you

can

dra

w th

is tr

iang

le, e

ither

. The

key

is ju

st to

dra

w o

ne.

§3

Rel

atio

nshi

ps b

etw

een

the

Trig

func

tions

– 1

21

Reca

ll th

at o

ne c

an d

ivid

e tw

o fra

ctio

ns b

y m

ultip

lyin

g by

the

reci

proc

al. T

hus

we

have

𝑎 𝑐∙𝑐 𝑏

.

But 𝑐 𝑐

is o

ne, a

nd th

eref

ore

the 𝑐

valu

es c

ance

l out

. Thi

s le

aves

us

with

𝑎 𝑏.

This

is a

ratio

! And

in fa

ct, it

is n

o m

ore

than

the

Tang

ent f

unct

ion’

s rat

io. T

hus w

e co

nclu

de

that

sin𝛼

cos𝛼

=ta

n𝛼

.

You

will

use

this

iden

tity

man

y, m

any

times

, and

so

we

form

aliz

e it

belo

w.

Quo

tient

iden

tity

tan𝛼

=si

n𝛼

cos𝛼

Exam

ple

5c

Sim

plify

co

s𝛼∙t

an𝛼

.

Follo

win

g ou

r rul

e of

thum

b, le

t’s c

onve

rt e

very

thin

g in

to S

ines

and

Cos

ines

. To

do th

is,

we

sim

ply

rena

me

tan𝛼

usi

ng o

ur id

entit

y ab

ove.

We

then

hav

e

cos𝛼∙

sin𝛼

cos𝛼

whe

nce

we

see

that

the

Cosin

es c

ance

l. Th

is s

impl

y le

aves

us

with

sin𝛼

.

This

proc

ess

of s

impl

ifyin

g ex

pres

sion

s w

ill b

e fo

rmal

ized

in U

nit s

ix. U

ntil

then

, the

goa

l is

to in

trod

uce

you

to th

is ki

nd o

f thi

nkin

g.

§𝟑 E

xerc

ises

1.)

Use

the

give

n Tr

ig ra

tio to

writ

e ou

t all

six

Trig

ratio

s in

eac

h pr

oble

m.

(A)

sin𝛼

=1

2

13

(B)

tan𝛼

=4 3

(C)

cos𝛼

=7 25

(D)

sin𝛼

=1 4

(E)

cos𝛼

=10

13

(F)

tan𝛼

=5 6

(G)

csc𝛼

=7 2

(H)

sec𝛼

=10 7

Uni

t fou

r – 1

22

2.)

Let u

s co

mpl

ete

Exam

ple

1c. I

f sin𝛼

=𝑎

, wha

t are

the

six

Trig

ratio

s?

3.)

Now

let t

an𝛼

=𝑑

. (A

) W

hat a

re th

e six

Trig

ratio

s gi

ven

this?

(B

) Co

mpa

re th

is an

swer

to th

e pr

evio

us.

4.)

In E

xam

ple

1c, w

e le

t th

e le

ngth

of t

he h

ypot

enus

e of

the

tria

ngle

to

equa

l 1. I

s th

at O

K? W

hy d

on’t

we

let

the

hypo

tenu

se e

qual

𝑐,

to a

llow

for

any

and

all

poss

ibili

ties?

Now

sup

pose

sin𝛼

=𝑥 𝑐, a

nd th

e le

ngth

s of

you

r tria

ngle

are

𝑎,𝑏

, and

𝑐,

with

the

hypo

tenu

se e

qual

ing 𝑐.

(A

) W

hat a

re th

e six

Trig

ratio

s?

(B)

Com

pare

this

with

the

prev

ious

two

resu

lts.

5.)

Giv

en th

e Tr

ig fu

nctio

ns a

nd a

ngle

mea

sure

, writ

e th

e eq

uiva

lent

cof

unct

ion.

(A

) si

n3

0°

(B)

cos

10

° (C

) co

t7

° (D

) se

c64

° (E

) co

s3

1°

(F)

tan

14

° (G

) cs

c4

7°

(H)

sin2

5°

(I)

tan( 𝛽

+𝛾)

(J)

sin𝛽

6.

) W

rite

out a

ll of

the

cofu

nctio

n id

entit

ies,

incl

udin

g th

e on

es w

e di

scov

ered

in th

e re

adin

g. H

int:

The

re a

re s

ix o

f the

m.

7.)

Now

writ

e ou

t all

of th

e re

cipr

ocal

iden

titie

s. H

int:

The

re a

re s

ix o

f the

m.

8.)

Eval

uate

the

follo

win

g.

(A)

sin

23

0°

(B)

cos2

30

° (C

) ta

n2

30

° (D

) co

s24

5°

(E)

tan

24

5°

(F)

sin

26

0°

(G)

cos2

60

° (H

) ta

n2

60

° (I)

si

n( 3

0°2)

(J)

cos2𝛼

9.

) W

rite

out a

tabl

e of

val

ues f

or s

in2𝛼

,co

s2𝛼

, and

tan

2𝛼

, sta

rtin

g at

𝛼=

0, g

oing

up

by 5

° eac

h ro

w, a

nd e

ndin

g at

𝛼=

90

°. Yo

u w

ill n

eed

a ca

lcul

ator

for t

his

Exer

cise

. 10

.) U

se y

our r

esul

ts fr

om th

e pr

evio

us E

xerc

ise to

ans

wer

the

follo

win

g qu

estio

ns.

(A)

Wha

t is

the

max

imum

val

ue o

f sin

2𝛼

,co

s2𝛼

, and

tan

2𝛼

? (B

) W

hat i

s th

e m

inim

um v

alue

of s

in2𝛼

,co

s2𝛼

, and

tan

2𝛼

? (C

) Ar

e th

ere

any

simila

ritie

s or

diff

eren

ces

betw

een

sin

2𝛼

and

sin𝛼

? Co

mpa

re

your

resu

lts fr

om th

e pr

evio

us s

ectio

n.

11.)

One

of

the

mos

t im

port

ant

rela

tions

hips

in

Trig

onom

etry

is

the

Pyth

agor

ean

Iden

tity

we

disc

usse

d in

the

read

ing.

Writ

e th

is id

entit

y do

wn

now

. 12

.) Ev

alua

te th

e fo

llow

ing.

(A

) si

n2

60

°+

cos2

60

° (B

) co

s23

0°

+si

n2

30

° (C

) si

n2(𝛼

2)

+co

s2(𝛼

2)

(D)

sin

2( 3𝛼

+𝜋)

+co

s2( 3𝛼

+𝜋)

13.)

It is

ofte

n he

lpfu

l to

rew

rite

sin

2𝛼

or c

os2𝛼

. Use

the

Pyth

agor

ean

Iden

tity

to re

writ

e si

n2𝛼

and

co

s2𝛼

. 14

.) Si

mpl

ify th

e fo

llow

ing.

§3

Rel

atio

nshi

ps b

etw

een

the

Trig

func

tions

– 1

23

(A)

tan𝛼∙c

sc𝛼

(B

) ( s

in𝛼

+co

s𝛼)2

15

.) Ar

e th

ere

any

othe

r Pyt

hago

rean

Iden

titie

s? T

o fin

d th

is o

ut, u

se a

cal

cula

tor a

nd

try

the

follo

win

g fo

r diff

eren

t val

ues

of 𝛼

. (A

) se

c2𝛼

+cs

c2𝛼

(B

) ta

n2𝛼

+co

t2𝛼

(C

) Th

ere

are

two

othe

r Py

thag

orea

n Id

entit

ies.

Firs

t, us

ing

the

prev

ious

tw

o,

gues

s w

hat t

hey

mig

ht b

e. T

hen,

if y

ou c

an’t

figur

e it

out,

look

them

up

and

writ

e th

em d

own

now

. We’

ll di

scov

er h

ow to

arr

ive

at th

ese

resu

lts w

hen

we

have

som

e be

tter

tool

s. 16

.) An

swer

Tru

e or

Fal

se.

(A)

sin𝛼

=co

s(𝛼−

90

°)

(B)

sin

2𝛼

+co

s2𝛽

=1

iff 𝛼

+𝛽

=9

0°

(C)

sin

2𝛼

=si

n𝛼∙𝛼

(D

) si

n2𝛼

is s

omet

imes

neg

ativ

e.vi

vi A

ssum

e 𝛼∈ℝ

.

 

Remote Learning Packet NB: Please keep all work produced this week. Details regarding how to turn in this work will be forthcoming.

April 27 - May 1, 2020 Course: Spanish III Teacher(s): Ms. Barrera anna.barrera@greatheartsirving.org Supplemental links: www.conjuguemos.com www.spanishdict.com Weekly Plan: Monday, April 27 ⬜ Capítulo 5 - Read an announcement for a job position then choose the appropriate answer.. ⬜ Capítulo 5 - Translate your answers. Tuesday, April 28 ⬜ Capítulo 5 - Read the curriculum of Ms. Diaz and then choose the appropriate answer.. ⬜ Capítulo 5 - Translate your answers. Wednesday, April 29 ⬜ Capítulo 5 - Read the curriculum of Mr. Perez and then choose the appropriate answer.. ⬜ Capítulo 5 - Translate your answers. Thursday, April 30 ⬜ Capítulo 5 - Read about the history of Spain from the Celtiberos to the conquest of the muslims by Fernando e Isabela. ⬜ Capítulo 5 - Second, match and complete with the appropriate vocabulary according to your comprehension of the reading. Friday, May 1 ⬜ Capítulo 5 - Vocabulary Quiz A and B ⬜ Capítulo 5 - Vocabulary Quiz - C Statement of Academic Honesty I affirm that the work completed from the packet is mine and that I completed it independently. _______________________________________Student Signature

I affirm that, to the best of my knowledge, my child completed this work independently _______________________________________ Parent Signature

Monday, April 27 Capítulo 5 - Read an announcement for a job position and then choose the appropriate answer. Then translate your answers for each number. A google document has been added in google classroom to type out your answers and submit within the lesson of the day. Please be organized and number your answers. I. Handout: Trabajo y comunidad: Lectura 1 - Read the Lectura and then choose the appropriate answer. II. Handout: Lectura 1 - Translate your answers. Tuesday, April 28 Capítulo 5 - Read Ms. Diaz’s Curriculum and then choose the appropriate answer. Then translate your answers for each number. A google document has been added in google classroom to type out your answers and submit within the lesson of the day. Please be organized and number your answers. I. Handout: Curriculum de Karen Díaz Manzanares: Lectura 2 - Read the Lectura and then choose the appropriate answer. II. Handout: Lectura 2 - Translate your answers. Wednesday, April 29 Capítulo 5 - Read the curriculum of Mr. Perez and then choose the appropriate answer. Then translate your answers for each number. A google document has been added in google classroom to type out your answers and submit within the lesson of the day. Please be organized and number your answers. I. Handout: Curriculum de Brandon Perez: Lectura 3 - Read the Lectura and then choose the appropriate answer. II. Handout: Lectura 3 - Translate your answers. III. Writing: Integration of ideas. Compare both curriculums. Then, according to your observations, then decide which of the two candidates is best prepared for the job from Lectura 1. Use information from the announcement and both curriculums to support your ideas. Thursday, April 30 Capítulo 5 - Read about the history of Spain from the Celtiberos to the conquest of the muslims by Fernando e Isabela. Second, match and complete with the appropriate vocabulary according to your comprehension of the reading. A google document has been added in google classroom to type out your answers and submit within the lesson of the day. Please when typing your answers label Exercise A, B, etc. I. Spanish Reading with Comprehension Exercises. La historia De España. Exercise A. Choose the appropriate letter. Exercise B. Choose the appropriate phrase in order to complete the sentence. Exercise C. Select the sentence that is false(falso) and rewrite it making it true. (cierto). For example, 1.

F then rewrite the sentence to make it true. Exercise D. Fill in the blank with the appropriate word from the reading. Friday, May 1 Capítulo 5 - Comprehension of work and volunteer vocabulary. I. Quiz 5-1: Comprensión del vocabulario 1. Type in your answers in the google doc provided for you in google classroom. A. Type in your 5 answers in the google doc. B. Type in your 8 answers in the google doc for this section. II. Quiz 5-1: Comprensión del vocabulario 1. Type in your answers in the google doc provided for you in google classroom for C. Because there are no numbers, please add them with your answers. There are 6 for the top and 6 for the bottom.

 

Physics Remote Learning Packet

May 4-8, 2020 Course: 11 Physics Teacher: Miss Weisse natalie.weisse@greatheartsirving.org Resource: Miss Weisse’s Own Physics Textbook — new pages found at the end of this packet Weekly Plan: Monday, May 4 ⬜ Read & Understand Notes on Unit 8 Part 3 – Hooke’s Law Background (pages 51-54) ⬜ Complete Pre-Lab Observations ⬜ Prepare Your “Lab Book” Entry Tuesday, May 5 ⬜ Take Data ⬜ Create 4 Graphs ⬜ Complete the Calculations and Analysis of All Four Graphs ⬜ Email Miss Weisse with Questions Wednesday, May 6 ⬜ Area Under the Curve Calculations for All Four Graphs ⬜ Explanation of the Physical Significance of the Area Under the Curve ⬜ Write Lab Conclusion ⬜ Email Miss Weisse with Questions and to Ask for Unit 8 Part 4 of Miss Weisse’s Own Physics Textbook Thursday, May 7 ⬜ Read & Understand Notes on Unit 8 Part 4 - Hooke’s Law & Elastic Energy Notes ⬜ Complete Unit 8 Worksheet 2 #1-4 ⬜ Email Miss Weisse with Questions and to Ask for Solutions Friday, May 8 ⬜ Attend Office Hours at 9:30am! ⬜ Review Unit 8 Part 4 - Hooke’s Law & Elastic Energy Notes (pages 55-57) ⬜ Complete Unit 8 Worksheet 2 #5-7 ⬜ Email Miss Weisse with Questions and to Ask for Solutions Statement of Academic Honesty I affirm that the work completed from the packet is mine and that I completed it independently. _______________________________________ Student Signature

I affirm that, to the best of my knowledge, my child completed this work independently _______________________________________ Parent Signature

Monday, May 4

➔ Read & Understand Notes on Unit 8 Part 3 – Hooke’s Law Background (pages 51-54)

➔ Complete Pre-Lab Observations

◆ Watch the short, two-minute video (https://safeYouTube.net/w/wMh8), observe the image of four spring scales below, and image the behavior of springs or rubber bands to answer the following three familiar questions on a sheet of paper, or on a google doc, to be turned in.

● What do you see?

● What can you measure?

● What can you change?

**As you observe the four spring scales, describe both the motion and behavior of the springs as they move and the difference between the four springs.

➔ Prepare Your “Lab Book” Entry on a piece of paper. This assignment cannot be completed online.

◆ Write the Purpose of the lab (can be found in Unit 8 – Part 3... if you don’t already know)

◆ List your Independent Variable (length of spring), Dependent Variable (force of spring), and Constants (determine these yourself).

*** Are our variable in your list of things we can measure and change?!

◆ Create four data charts like the example chart below.

NOTE! Extension is the length the spring stretched when a force is applied, not the total length of the spring. To calculate this, subtract the length at 0 Newtons force from each measure length.

Extension = Current Length – Length @ 0 Newton Force

Green Scale Force (N) Length (cm) Extension (cm) Extension (m)

Tuesday, May 5

➔ Take data using the images found at the end of this packet, after the textbook.

◆ Don’t forget our precision measurement rules! You should write a measurement that has one more decimal place than the ruler marks. So each of your measurements should have 2 decimal places. Also, to take good data, zoom in on your screen to ~200%, then use a piece of paper to create a line between the bottom of the spring and the ruler markings.

➔ Create four graphs from your four sets of data. Each graph should take up at least ⅓ - ½ of the page.

➔ Complete the calculations and analysis for each of the four graphs

As a reminder –

◆ The calculations include the slope, the y-intercept, and the 5% rule for the y-intercept if your value is not zero.

◆ The analysis includes what the slope tells us (focus on units!) and what the y-intercept tells us.

◆ Some helpful reminders are found on page 54 of Miss Weisse’s Own Physics Textbook

Wednesday, May 6

➔ Now that you have an equation that describes the relationship between force exerted by the spring and the extension of the spring, it is time to relate this information to ENERGY. If you recall from our study of motion 1st semester, we found new, important physical properties by calculating the Area Under the Curve (AUC) of the velocity versus time graph and acceleration vs. time graph. We can do the same thing here with our spring force versus length graph. Calculate the AUC for all four graphs.

➔ Explain the physics significance of the Area Under the Curve. What physical quantity does this measurement give us? Use the units to help puzzle it out.

➔ Write a Lab Conclusion.

◆ Restate the problem and summarize how the data was taken. ◆ State the relationship between the two variables, explain what the slope means, and state what the

y-intercept tells us. ◆ Compare the four different springs, giving numerical data. ◆ Explain what the area under the curve tells us. ◆ If your data has problems, provide possible explanations. ◆ Provide an answer to the original problem.

***Today’s work can be done on the same piece of paper as Tuesday’s lab work.

➔ Email Miss Weisse with Questions and to Ask for Unit 8 Part 4 of Miss Weisse’s Own Physics Textbook

Thursday, May 7

➔ If you have not done so yet, turn in your lab and request Unit 8 Part 4 from Miss Weisse via Email!

➔ Read & Understand Notes on Unit 8 Part 4 - Hooke’s Law & Elastic Energy

➔ On a sheet of paper with a full heading, complete Unit 8 Worksheet 2 #1-4

Energy Storage and Transfer Model Worksheet 2 (part #1-4): Hooke’s Law and Elastic Energy

Directions: Suppose one lab group found that F = 1000 N/m (∆x). Construct a graphical representation of force vs. displacement. (Hint: make the maximum displacement 0.25 m. )

1. Graphically determine the amount of energy stored while stretching the spring described above from x = 0 to x = 10. cm. 2. Graphically determine the amount of energy stored while stretching the spring described above from x = 15 to x = 25 cm.

Directions: The graph below was made from data collected during an investigation of the relationship

between the amounts two different springs stretched when different forces were applied. 3. Determine the spring constant for each spring.

4. For each spring, compare: a. the amount of force required to stretch the spring 3.0 m. b. the Eel stored in each spring when stretched 3.0m.

Friday, May 8

➔ Attend Office Hours at 9:30am! I want to talk to you and help you!

➔ Review Unit 8 Part 4 - Hooke’s Law & Elastic Energy Notes

➔ On a sheet of paper with a full heading, complete Unit 8 Worksheet 2 #5-7

Energy Storage and Transfer Model Worksheet 2 (#5-7): Hooke’s Law and Elastic Energy

5. Determine the amount that spring 2 needs to be stretched in order to store 24 joules of energy. 6. The spring below has a spring constant of 10. N/m. If the block is pulled 0.30 m horizontally to the

right, and held motionless, what force does the spring exert on the block? Sketch a force diagram for the mass as you hold it still. (Assume a frictionless surface.)

7. The spring below has a spring constant of 20. N/m. The µs between the box and the surface is 0.40.

a. The box is pushed to the right, then released. Draw a force diagram for the box above when the spring

is stretched, yet the box is stationary. b. What is the maximum distance that the spring can be stretched from equilibrium before the box begins

to slide back? c. Do pie chart analysis for this situation, when the spring is stretched beyond its maximum (from part b

above) so it slides back, and then the box oscillates back and forth until it comes to a stop. Assume your system includes the spring, box, and table top.

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SET UP ********************to make your measurements you will want to zoom in on your computer screen (~200%)********************

FROM THE FRONT: The spring scale is being anchored in place with a pen perpendicular to the board, so the whole scale cannot be pulled forward when the hook is pulled forward to extend the spring.

FROM THE TOP: Again, the spring scale is anchored by the pen and the top of the spring is aligned with the zero mark of the ruler. The ruler remains stationary throughout the measurements.

GREEN SPRING SCALE (VIDEO)

WHITE SPRING SCALE (VIDEO)

RED SPRING SCALE (VIDEO)

BROWN SPRING SCALE (VIDEO)