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2D 5 day 3 Primary Trig Ratios.notebook 1 December 07, 2016 Primary Trigonometric Ratios Learning Goal: Learn and use the primary trig ratios Activity B H F D A C I G E

Primary Trigonometric Ratios - MPM 2D · Primary Trigonometric Ratios Sine of LA Cosine of LA Tangent of LA _Þ opposite sin A hypotenuse adiacent cos A hypotenuse opposite tan A

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  • 2D  5  day 3  Primary Trig Ratios.notebook

    1

    December 07, 2016

    Primary Trigonometric Ratios

    Learning Goal: Learn and use the primary trig ratios

    Activity

    B H F DA

    C

    I

    G

    E

  • 2D  5  day 3  Primary Trig Ratios.notebook

    2

    December 07, 2016

    Primary Trig Ratios Powerpoint

  • 2D  5  day 3  Primary Trig Ratios.notebook

    3

    December 07, 2016

    Using your calculator...

    1.  Make sure it is set to DEGREEs.

    sin 90o = 1

    2.  Going forward...solve

    sin 30o =

    sin 40o =

    sin 120o =

    sin 240o =

  • 2D  5  day 3  Primary Trig Ratios.notebook

    4

    December 07, 2016

    3.  Going backwards...find the angle

    sin θ = 1 sin θ = 0.5

    cos θ = 0.5 cos θ = 2

    Soh Cah Toa

    x

    15 cm

    35o

    Steps:

    1. Name the sides2. Choose equation3. Substitute in numbers4. Solve

  • 2D  5  day 3  Primary Trig Ratios.notebook

    5

    December 07, 2016

    On the Boards...

    Find x.

    x

    15 cm

    23 cm

  • 2D  5  day 3  Primary Trig Ratios.notebook

    6

    December 07, 2016

    Homework

    pg. 398 # 1-3, 5, 6, 10, 13

  • Attachments

    MidptCoordPlane.tns

    MidptCoordPlane_Student.pdf

    MidptCoordPlane_Teacher.doc

    MidptCoordPlane_Student.doc

    Midpoint.tns

    Ch05.070trigratios_10Ac.ppt

    SMART Notebook

  • Midpoints in the Coordinate Plane MidptCoordPlane.tns

    Name ________________________

    Class ________________________

    ©2010 Texas Instruments Incorporated Page 1 Midpoints in the Coordinate Plane

    Problem 1 – Midpoints of Horizontal or Vertical Segments

    On page 1.3, construct a horizontal segment and a vertical segment. Find the coordinates of the endpoints and then predict the coordinates of the midpoints of the segments.

    Endpoints Predicted Midpoint

    (_____ , _____) and (_____ , _____) (_____ , _____)

    (_____ , _____) and (_____ , _____) (_____ , _____)

    Describe how you can predict the coordinates of the midpoint of a horizontal or vertical segment.

    Problem 2 – Midpoints of Diagonal Segments

    On page 2.2, construct two diagonal segments. Find the coordinates of the endpoints and then make a predication about the coordinates of the midpoints.

    Endpoints Predicted Midpoint

    (_____ , _____) and (_____ , _____) (_____ , _____)

    (_____ , _____) and (_____ , _____) (_____ , _____)

    Describe how you can predict the coordinates of the midpoint of a diagonal segment.

    Apply The Math

    What formula gives the midpoint of a segment with endpoints (x1, y1) and (x2, y2)?

  • Midpoints in the Coordinate Plane

    ©2010 Texas Instruments Incorporated Page 2 Midpoints in the Coordinate Plane

    Determine the midpoint of a segment with the following endpoints:

    1. (3, 10) and (5, 10)

    2. (1, 8) and (8, 9)

    3. (7, 2) and (4, 4)

    4. (–2, 3) and (5, –7)

    5. (1.8, 4.9) and (7.2, 2.7)

    6. (–3.3, 5.5) and (–5.5, 3.3)

    Given an endpoint and midpoint of a segment, find the other endpoint:

    7. Endpoint: (3, 1); Midpoint: (3, 4)

    8. Endpoint: (2, 5); Midpoint: (5, 6)

    9. Endpoint: (–4, 3); Midpoint: (1, 0)

    Extension – Trisection Points

    On page 3.2, segment PQ has two trisection points, which divide PQ into 3 equal parts. Drag P or Q to change the segments location. Find the coordinates of the endpoints and then make a prediction about the coordinates of the trisection points.

    Endpoints Predicted Trisection points

    (_____ , _____) and (_____ , _____) (_____ , _____) and (_____ , _____)

    (_____ , _____) and (_____ , _____) (_____ , _____) and (_____ , _____)

    Describe how you can predict the coordinates of the trisection points of a segment.

    SMART Notebook

    TImath.com Geometry

    TImath.com Geometry

    Midpoints in the Coordinate Plane

    ID: 8612

    Time required

    40 minutes

    Activity Overview

    In this activity, students will explore midpoints in the coordinate plane. Beginning with horizontal or vertical segments, students will show the coordinates of the endpoints and make a conjecture about the coordinates of the midpoint. This conclusion is extended to other segments in the coordinate plane.

    Topic: Points, Lines & Planes

    · Given the coordinates of the ends of a line segment, write the coordinates of its midpoint.

    Teacher Preparation and Notes

    · This activity is intended to be used in a middle school or high school geometry classroom.

    · This activity is designed to be student-centered with the teacher acting as a facilitator while students work cooperatively. Use the following pages as a framework as to how the activity will progress.

    · Depending on student skill level, you may wish to use points with integer coordinates, or only positive values.

    · The Coordinate Midpoint formula for the midpoint of (x1, y1) and (x2, y2) is

    1212

    xxyy

    ,

    22

    ++

    æö

    ç÷

    èø

    . This can also be expressed as “The coordinates of the midpoint of a line segment are the averages of the coordinates of the endpoints.”

    · Trisection Points is an optional extension, which can be used depending on time and student ability. The trisection points of a segment with endpoints (x1, y1) and (x2, y2) are

    1212

    xxyy

    ,

    33

    ++

    æö

    ç÷

    èø

    and

    1212

    2(xx)2(yy)

    ,

    33

    ++

    æö

    ç÷

    èø

    .

    · To download the student TI-Nspire document (.tns file) and student worksheet, go to http://education.ti.com/exchange and enter “8612” in the search box.

    Associated Materials

    · MidPtCoordPlane_Student.doc

    · MidPtCoordPlane.tns

    Suggested Related Activities

    To download any activity listed, go to education.ti.com/exchange and enter the number in the quick search box.

    · Division of Integers (TI-84 family) — 1433

    · Multiplication of Integers (TI-84 family) — 1434

    · Integers (TI-84 family, TI-Navigator Technology) — 4412

    Problem 1 – Midpoints of Horizontal or Vertical Segments

    Have students open the file and read the directions on page 1.2.

    On page 1.3, students are to construct a horizontal segment in the first quadrant. Using the Point On tool (MENU > Points & Lines > Point On) they need to plot two points directly on grid points and then connect the points using the Segment tool (MENU > Points & Lines > Segment).

    Direct students to show the coordinates for the endpoints of the segment using the Coordinates and Equations tool (MENU > Tools > Coordinates and Equations).

    Note: Press x (or ·) once to select the point whose coordinates you wish to show and then press x (or ·) again to anchor the measurement.

    Students should make a prediction about the coordinates for the midpoint of the segment.

    To check their predictions, have them select MENU > Construction > Midpoint, construct the midpoint of the segment, and show the coordinates of the midpoint using the Coordinates and Equations tool.

    Tell students to hide the coordinates of the midpoint with the Hide/Show tool (MENU > Tools > Hide/Show) before moving the segment.

    Students can now drag the endpoints to create a vertical segment and then make a prediction for the new coordinates of the midpoint. They can verify their prediction by using the Hide/Show tool again.

    If desired, have students explore the midpoint of a segment whose endpoints do not have integer coordinates by selecting MENU > Window > Window Settings and dividing each value by 10. They can also explore what happens when one or both endpoints are not in Quadrant 1.

    Problem 2 – Midpoints of Diagonal Segments

    Direct students to advance to page 2.1 and read the directions.

    On page 2.2, students are to use the Point On tool and the Segment tool to construct a diagonal segment in the first quadrant.

    Students once again need to use the Coordinates and Equations tool to show the coordinates for the endpoints of the segment.

    They should use the endpoint coordinates to make a predication about the coordinates of the midpoint.

    To check their prediction, students can construct the midpoint and then find the coordinates.

    Tell students hide the coordinates of the midpoint with the Hide/Show tool (MENU > Tools > Hide/Show).

    Students can drag the endpoints to create a different diagonal segment and then make a prediction for the new coordinates of the midpoint. Students can confirm their prediction by using the Hide/Show tool again.

    Discuss how to find the coordinates of the midpoint of a segment if the coordinates of the endpoints are known. Then, challenge students to write a formula or a rule for calculating midpoints.

    To observe how the midpoint is related to the endpoints, students should press MENU > Tools > Text and enter the expression (a+b)/2 in a text box.

    Then select MENU > Tools > Calculate, click on the expression, and click on the x‑values of the coordinates for the endpoints for a and b. Tell students to repeat the calculation with the y‑values of the coordinates for the endpoints.

    Now they can drag the segment endpoints and observe the calculation results as they updates.

    Discuss how these calculation results relate to the coordinates of the midpoint.

    Extension – Trisection Points

    Students will read the directions on page 3.1 and then advance to page 3.2. There is a segment displayed with its two trisection points, which divide the segment into three equal sections.

    If needed, the can select MENU > Measurement > Length and measure the lengths of each section to confirm that the segment is trisected.

    The Coordinates and Equations tool should be used by students to show the coordinates of the segment endpoints.

    Students should make a prediction about the coordinates of the trisection points. To confirm their prediction, students can show the coordinates of the trisection points.

    Have students explore the relationship between the coordinates of the segment endpoints and the trisection points by dragging points P or Q to adjust the location of the segment. Challenge them to write a formula for the trisection points.

    ©2010 Texas Instruments IncorporatedTeacher PageMidpoints in the Coordinate Plane

    ©2010 Texas Instruments IncorporatedPage 2Midpoints in the Coordinate Plane

    _1250919735.unknown

    _1250919749.unknown

    _1250919717.unknown

    SMART Notebook

    Midpoints in the Coordinate Plane

    MidptCoordPlane.tns

    Name ________________________

    Class ________________________

    Midpoints in the Coordinate Plane

    Problem 1 – Midpoints of Horizontal or Vertical Segments

    On page 1.3, construct a horizontal segment and a vertical segment. Find the coordinates of the endpoints and then predict the coordinates of the midpoints of the segments.

    Endpoints Predicted Midpoint

    (_____ , _____) and (_____ , _____) (_____ , _____)

    (_____ , _____) and (_____ , _____) (_____ , _____)

    Describe how you can predict the coordinates of the midpoint of a horizontal or vertical segment.

    Problem 2 – Midpoints of Diagonal Segments

    On page 2.2, construct two diagonal segments. Find the coordinates of the endpoints and then make a predication about the coordinates of the midpoints.

    Endpoints Predicted Midpoint

    (_____ , _____) and (_____ , _____) (_____ , _____)

    (_____ , _____) and (_____ , _____) (_____ , _____)

    Describe how you can predict the coordinates of the midpoint of a diagonal segment.

    Apply The Math

    What formula gives the midpoint of a segment with endpoints (x1, y1) and (x2, y2)?

    Determine the midpoint of a segment with the following endpoints:

    1. (3, 10) and (5, 10)

    2. (1, 8) and (8, 9)

    3. (7, 2) and (4, 4)

    4. (–2, 3) and (5, –7)

    5. (1.8, 4.9) and (7.2, 2.7)

    6. (–3.3, 5.5) and (–5.5, 3.3)

    Given an endpoint and midpoint of a segment, find the other endpoint:

    7. Endpoint: (3, 1); Midpoint: (3, 4)

    8. Endpoint: (2, 5); Midpoint: (5, 6)

    9. Endpoint: (–4, 3); Midpoint: (1, 0)

    Extension – Trisection Points

    On page 3.2, segment PQ has two trisection points, which divide

    PQ

    into 3 equal parts. Drag P or Q to change the segments location. Find the coordinates of the endpoints and then make a prediction about the coordinates of the trisection points.

    Endpoints Predicted Trisection points

    (_____ , _____) and (_____ , _____) (_____ , _____) and (_____ , _____)

    (_____ , _____) and (_____ , _____) (_____ , _____) and (_____ , _____)

    Describe how you can predict the coordinates of the trisection points of a segment.

    ©2010 Texas Instruments IncorporatedPage 1Midpoints in the Coordinate Plane

    ©2010 Texas Instruments IncorporatedPage 2Midpoints in the Coordinate Plane

    _1250933532.unknown

    SMART Notebook

    SMART Notebook

    www.thevisualclassroom.com

    Trigonometry: The study of triangles (sides and angles)

    physics

    surveying

    Trigonometry has been used for centuries in the study of:

    astronomy

    geography

    engineering

    www.thevisualclassroom.com

    A

    B

    C

    adjacent

    hypotenuse

    opposite

    www.thevisualclassroom.com

    A

    B

    C

    adjacent

    opposite

    hypotenuse

    www.thevisualclassroom.com

    A

    B

    C

    opposite

    adjacent

    hypotenuse

    www.thevisualclassroom.com

    A

    B

    C

    opposite

    adjacent

    hypotenuse

    www.thevisualclassroom.com

    A

    B

    C

    opp

    adj

    hyp

    SOH

    CAH

    TOA

    www.thevisualclassroom.com

    A

    B

    C

    8

    6

    10

    opp

    adj

    hyp

    SOH

    CAH

    TOA

    www.thevisualclassroom.com

    A

    B

    C

    3

    4

    5

    adj

    opp

    hyp

    SOH

    CAH

    TOA

    www.thevisualclassroom.com

    A

    B

    C

    5

    12

    13

    adj

    opp

    hyp

    SOH

    CAH

    TOA

    www.thevisualclassroom.com

    sin 21° =

    cos 53° =

    tan 72° =

    0.3584

    0.6018

    3.0777

    Use a calculator to determine the following ratios.

    www.thevisualclassroom.com

    sin A = 0.4142

    cos B = 0.6820

    tan C = 1.562

    ÐA = sin-1(0.4142)

    ÐB = cos-1(0.6820)

    ÐC = tan-1(1.562)

    = 24°

    = 47°

    = 57°

    Determine the following angles (nearest degree).

    www.thevisualclassroom.com

    Determine the following angles (nearest degree).

    sin A =

    cos B =

    tan C =

    ÐA = sin-1(0.5833)

    ÐB = cos-1(0.2666)

    ÐC = tan-1(1.875)

    = 36°

    = 75°

    = 62°

    = 0.5833

    = 0.2666

    = 1.875

    www.thevisualclassroom.com

    A

    B

    C

    a

    6 cm

    Example 1: Determine side a

    30º

    a = 6 sin 30°

    a = 3 cm

    a = 6 (0.5)

    hyp

    opp

    SOH

    CAH

    TOA

    www.thevisualclassroom.com

    A

    B

    C

    50º

    b

    9 m

    40º

    Ex. 2: Name two trig ratios that will allow us to calculate side b.

    www.thevisualclassroom.com

    A

    B

    C

    Example 3: Determine side b

    55º

    b

    8 cm

    b = 8 tan 55°

    b = 11.4 cm

    b = 8 (1.428)

    opp

    adj

    SOH

    CAH

    TOA

    www.thevisualclassroom.com

    P

    Q

    R

    12 cm

    17 cm

    Example 4: Determine the measure of ÐP.

    cos P = 0.70588

    ÐP = 45.1°

    ÐP = cos–1(0.70588)

    adjacent

    hypotenuse

    SOH

    CAH

    TOA

    www.thevisualclassroom.com

    P

    Q

    R

    q

    12 cm

    Example 5: Determine the measure of side PR.

    q(tan 35°) = 12

    q = 17.1 cm

    35°

    opp

    adj

    Method 1

    www.thevisualclassroom.com

    P

    Q

    R

    q

    12 cm

    Example 6: Determine the measure of side PR.

    q = 12(tan 55°)

    q = 17.1 cm

    35°

    opp

    adj

    Method 2

    ÐQ = 90° – 35°

    ÐQ = 55°

    55°

    q = 12(1.428)

    www.thevisualclassroom.com

    Ex. 7: In DPQR, ÐQ = 90°.

    a) Find sin R if PR = 8 cm and PQ = 4 cm.

    4 cm

    8 cm

    b) Find cos R .

    RQ2 = 82 – 42

    RQ2 = 64 – 16

    RQ2 = 48

    R

    P

    Q

    opp

    sin

    hyp

    B

    =

    adj

    cos

    hyp

    B

    =

    opp

    tan

    adj

    B

    =

    sin

    6

    A

    a

    =

    cos

    B

    =

    3

    5

    4

    3

    48

    RQ

    =

    opp

    sin

    hyp

    A

    =

    adj

    cos

    hyp

    A

    =

    12

    cos

    17

    P

    =

    sin30

    6

    a

    =

    o

    opp

    tan

    adj

    A

    =

    2)sin50

    9

    b

    =

    o

    6.9

    RQ

    =

    cos

    A

    =

    tan

    A

    =

    4

    5

    tan

    B

    =

    tan

    8

    B

    b

    =

    12

    tan35

    q

    =

    o

    12

    tan35

    q

    =

    o

    12

    0.7007

    q

    =

    tan55

    8

    b

    =

    o

    sin

    A

    =

    1)cos40

    9

    b

    =

    o

    1

    2

    =

    4

    sin

    8

    R

    =

    7

    12

    6.93

    cos

    8

    R

    =

    30

    R

    Ð=°

    4

    15

    15

    8

    tan55

    12

    q

    =

    o

    sin

    B

    =

    12

    13

    5

    13

    12

    5

    8

    10

    6

    10

    8

    6

    cos0.87

    R

    =

    SMART Notebook

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