The Six Trigonometric Ratios

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III. PRE ASSESSMENTLet us find out how much you already know about this module. Answer the following ques-tions as much as you can by writing on your answer sheet the letter that you think is the correctanswer. Take note of the items that you were not able to answer correctly and then let us find

out the correct answer as we go through this module.1. With respect to the given angle, what is the ratio of the hypotenuse to the opposite side?

A. sine B. cosine C. tangent D. cosecant

2. Which of the following statements is correct?

A. x = 8 C. sin 60° = y 4

B. sin30° =1 x

D. cos 60° =4 y

3. Given the figure on the right, which refers to the angle of depression?

A. ∠MKN C. ∠LKNB. ∠MKL D. none of these

4. In the triangle PQR, what is the length of PQ?A. 1 cm C. 9 cmB. 5 cm D. 12 cm

5. In the triangle DEF, what is m ∠E to the nearest degree?A. 16° C. 41°B. 28° D. 62°

17

15D

F

E

12 cm

P

Q R

13 cm

object

objectN

M

L

Keye

x

y 60 °

4


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6. If p = 30 and q = 60, what is the measure of ∠R?A. 40° C. 60°B. 55° D. 65°

7. Find the value of side c to the nearest unit.A. 20 C. 23B. D. 24

8. What is the value of x ?A. 3 C. 6

B. 3 2 D. 6 2

9. The expression 2(sin 30°) – tan 45° is equal to

A. 0 B. 1 C. 2 D. –1

10. A kite held by 125 m of string makes an angle of elevation with the ground of 45°. Abouthow high is the kite above the ground?A. 62.8 m C. 88.4 m

B. 75.1 m D. 113.6 m11. From the top of a barn 7.62 m high, you see a cat on the ground. The angle of depression of

the cat is 40°. How many meters must the cat walk to reach the barn?A. 9.08 m C. 9.81 mB. 9.80 m D. 9.18m

12. ABCD is a parallelogram. If AB is 8 cm long, BC is 5 cm and their included angle measures100°, how long is diagonal AC?A. 12.95 cm C. 10.40 cm

B. 12.59 cm D. 10.14 cm

13. In right triangle PQR, PQ = 12 cm and QR = 5 cm. What is cos R?

A. 1213

B. 513

C. 512

D. 125

x 45 ° 6

45 °

40 °

c

15

b

Q R

P

q

p

r


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14. ∆ XYZ is a non – right triangle. If XY measures 20 cm, XZ measures 15 cm and ∠Z measures35° then what is the measure of ∠Y?

A. 25.84° B. 24.85° C. 25.48 D. 24.58°

15. A balloon is 50 m high. Its angle of elevation from observer A is 45° and from observer B itis 30°. What is the maximum distance between the two observers? Express your answer tothe nearest meter.

A. 136 m B. 137 m C. 138 m D. 39 m

16. With the sun, a girl 1.4 m tall casts a 3.6 m shadow. Find the angle of elevation from the tipof the shadow to the sun. Express your answer to the nearest degree.

A. 19° B. 20° C. 21° D. 22°

17. You are walking along a straight level path toward a mountain. At one point the angle ofelevation of the top of the mountain is 40°. As you walk 250 m closer, the angle of elevationis 45°. How high is the mountain? Express your answer to the nearest tenth of a meter.

A. 1304 m B. 1340 m C. 1034 m D. 1043 m

18. Two points P and Q on the same side of a river are 12 m apart. A tree on the opposite side ofthe river is directly opposite a point between P and Q. The lines of sight of a tree across theriver make angles of 78° and 57°, respectively, with the line joining P and Q. Find the dis-tance from point Q to the tree. Express your answer to the nearest meter.

A. 14 m B. 15 m C. 16 m D. 17 m

19. A surveyor sights two signs and the angle between the two lines of sight measures 55°. If thefirst and the second signs are 70 m and 50 m away, respectively, from the surveyor, find thedistance between the two signs. Express your answer to the nearest meter.

A. 81 m B. 82 m C. 83 m D. 84 m

20. Two ships leave the same port at the same time. One ship sails on a course of 110° at 32 mi/h.The other sails on a course of 230° at 40 mi/h. Find the distance between them after 2 hours.Express your answer to the nearest mile.

A. 124 mi B. 125 mi C. 126 mi D. 127 mi

IV. LEARNING GOALSThe learner demonstrates understanding of the basic concepts of trigonometry and is able toapply the concepts of trigonometric ratios to formulate and solve real-life problems with pre-cision and accuracy.


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The Six Trigonometric Ratios: Sine, Cosine, Tangent, Secant, Cosecant, and Cotangent

What to KNOW

Lesson 1 will help you recall the different concepts about triangles. This will guide you todefine and illustrate the six trigonometric ratios.

➤ Activity 1: Triangles of Different SizesThis activity helps you recall the concepts of similar triangles.Investigate the following triangles:1. Draw three similar right triangles ABC, DEF, and GHI in different sizes in such a way that

m ∠C = m ∠F = m ∠ I = 63°.2. Measure the second acute angle in each of the triangles.3. Use a ruler to measure the sides of the triangles to the nearest tenths in centimeters. Then

find each of the following ratios for all the three triangles:a. the ratio of the leg opposite the 63° angle to the hypotenuseb. the ratio of the leg adjacent to the 63° angle to the hypotenuseRecord your findings in the given table

Measures in ∆ ABC in∆ DEF in ∆ GHI

leg opposite the 63° angle

leg adjacent to the 63° angle

hypotenuse

leg opposite 63 ° anglehypotenuse

leg a jacent to 63 ° anglehypotenuse

leg opposite 63 ° angle

leg adjacent to 63 ° angle

4. Analyze the ratios of the sides across these triangles. What do you notice about:a. the first ratio, leg opposite 63° angle to the hypotenuse?b. the second ratio, leg adjacent to 63° angle to the hypotenuse?c. the third ratio, leg opposite the 63° angle to the leg adjacent to 63° angle?

1


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5. Using a scientific calculator, determine the value of sin 63°, cos 63°, and tan 63° thencompare them to the values obtained in step 4. What do you observe?

6. Complete each statement:The ratio _____ is the approximate value of sine of 63°.

The ratio _____ is the approximate value of cosine of 63°.The ratio _____ is the approximate value of tangent of 63°.

7. Complete the sentence:In a right triangle having an acute angle,

a. the sine θ is the ratio between _________and ___________.b. the cosine θ is the ratio between ___________ and ___________.c. the tangent θ is the ratio between __________and ____________.

From Activity 1, you have discovered the different ratios derived from the sides of a righttriangle having an acute angle. The next activity will help you further develop the concept you have determined.

➤ Activity 2: Measuring and CalculatingFor each of the right-angled triangles in the worksheet do the following:1. Measure the marked angles;2. Label the opposite and adjacent sides with respect to the marked angle;

3. Use your calculator to work out the three ratios: OppositeHypotenuse

, AdjacentHypotenuse

, OppositeAdjacent

4. Record the measurements and calculations in this table.

Length of sides (in cm) Trigonometric ratios

Angle ( °) Opposite Adjacent Hypotenuse

AdjacentHypotenuse

OppositeHypotenuse

OppositeAdjacent

a)

b)

c)

d)

e)

f)


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5. Examine the values of the three ratios. Could the ratiosOpposite

Hypotenuse or Adjacent

Hypotenuse be greater than

1? Explain your answer.

6. What about the ratioOppositeAdjacent

? What greatest value could this ratio have? Why do you say so?

WORKSHEET

a. b.

c. d.

e. f.


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To summarize what you have learned from the activities, study the concepts below aboutthe Six Trigonometric Ratios.

In a right triangle, we can define actually six trigonometric ratios. Consider the right triangleABC below. In this triangle we let θ represent - ∠B. Then the leg denoted by a is the side adjacentto θ , and the leg denoted by b is the side opposite to θ .

We will use the convention that angles are symbolized by capital letters, while the sideopposite each angle will carry the same letter symbol, in lowercase.

sin of θ = sin θ =opposite

hypotenusecosecant of θ = csc θ =

hypotenuseopposite

cosine of θ = cos θ =adjacent

hypotenusesecant of θ = sec θ =

hypotenuseadjacent

tangent of θ = tan θ =oppositeadjacent

cotangent of θ = cot θ =adjacentopposite

SOH – CAH – TOA is a mnemonic usedfor remembering the equations.

Notice that the three new ratios at theright are reciprocals of the ratios on the left.Applying algebra shows the connectionbetween these functions.

csc θ =hypotenuse

opposite= 1

oppositehypotenuse

= 1sin θ

sec θ = hypotenuseadjacent

= 1adjacent

hypotenuse

= 1cos θ

cot θ =adjacentopposite

= 1oppositeadjacent

= 1tan θ

A

C B

θ

a

b c

hypotenuse

adjacent

opposite


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Showing a formula for the Missing Parts of a Right TriangleExample 1 . Determine the equation or formula to find a missing part of the triangle.

a. Solve for s in the figure above. b. Solve for p in the figure above.Solution :∠P is an acute angle, t is the Solution: ∠P is an acute angle, t is thehypotenuse, s is the side adjacent to ∠P. hypotenuse, and p is the opposite side of ∠P.Use CAH, that is Use SOH, that is

cos θ =adjacent

hypotenuse

cos P =st

cos 53° =st

s = 12 cos 53°

sin θ =opposite

hypotenuse

sin P = pt

sin 53° = p12

p = 12 sin 53°

c. Solve for a in figure 2.Solution: ∠B is an acute angle, b is the opposite side, and a is the side adjacentto ∠B. Use TOA, that is

tan θ =oppositeadjacent

= ba

tan B =10.6

a

tan 67° =10.6

aa tan 67° = 10.6

a =10.6

tan 67°

T

p

S

P s

t = 12

53°

B

C A

c a

b = 10.6

67°


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d. Solve for c in figure 2.Solution : ∠B is an acute angle, b is the opposite side and c is the hypotenuse of the givenacute angle. Use SOH, that is

sin c =opposite

hypotenusec sin 67° = 10.6

sin B =bc

c =10.6

sin 67°

sin 67° =10.6

c

Using the Calculator to Find Trigonometric RatiosAs you saw in previous sections, the values of the trigonometric ratios for any particular angleare constant, regardless of the length of the sides. These values can also be found using a calcu-lator. Also, you can use a calculator to find an angle when you are given a trigonometric ratio.

a. Finding a ratio given the angleExample: To find the value of sin 38°, ensure that your calculator is operating in degrees.Solution:

Press sin 38 = 0.615661475

The calculator should give sin 38° = 0.616, correct to three decimal places.

b. Finding an angle given the ratioIn finding the size of the angle to the nearest minute, given the value of the trigonometricratio, just follow the steps in the examples below.Example: sin θ = 0.725, find θ to the nearest minuteSolution:

Press 2ndF sin 0.725 = 46.46884783

To convert this to degrees/minutes/seconds mode,

Press 2ndF D°M’S

The calculator gives you 46° 28’ (nearest minute)


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c. Degrees and minutesSo far, all angles have been in whole degrees. However, one degree can be divided equallyinto 60 minutes. Further, one minute can be divided equally into 60 seconds. Angle meas-urement can also be expressed in degree/minutes form. We can use a calculator to convert adegree measure from decimal form to degree, minute, and second form.Example: Write 54.46° in degree and minutes, giving an answer correct to the nearest minute.Solution:

Press 54.46° 2ndF D°M’S

The calculator gives 54°27’36”, or 54°28’. (nearest minute)

Try This

1. Use your calculator to find the value of the following, correct to two decimal places.a. cos 85° d. cos 65°b. sin 7° e. tan 23°c. tan 35°

2. Using the degrees/minutes/seconds button on your calculator, write each of the followingin degrees and minutes, give answers to the nearest minute.a. 17.8° d. 108.33°b. 48.52° e. 35.24°

c. 63.7°

3. Find the size of the angle θ (to the nearest degree) where θ is acute.a. sin θ = 0.529 d. sin θ = 0.256b. cos θ = 0.493 e. tan θ = 0.725c. tan θ = 1.8

4. Find m ∠ θ , to the nearest minute, given that θ is acute.a. sin θ = 0.9 d. cos θ = 0.501

b. cosθ

= 0.013 e. tanθ

= 2.3c. tan θ = 0.958


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A. Solving a right triangle given the measure of the two parts; the length of thehypotenuse and the length of one leg

Solving a right triangle means finding the measure of the remaining parts.

Example:Triangle BCA is right-angled at C. If c = 23 and b = 17, find ∠ A, ∠ B and a . Express youranswers up to two decimal places.Solution: Sketch a figure:a. Side b is the adjacent side of ∠A; c is the hypotenuse

of right triangle BCA. Use CAH, that is

cos θ =adjacent

hypotenuse

cos A =b

c cos A =

1723

cos A = 0.7391

We can use our scientific calculator to find an angle whose cosine value is 0.7391.Using a scientific calculator, A = 42.340°

b. Since in part (a), it was already found that ∠A = 42.34°, then ∠B = 90° – 42.34°

∠B = 47.66°.c. Using the Pythagorean theorem:

a2 + b2 = c2

a2 + (17) 2 = (23) 2

a2 + 289 = 529 a2 = 529 -289 a2 = 240

a = 240 a = 15.49

B. Solving a Right Triangle Given the Length of the Hypotenuse and the Measure ofOne Acute Angle

Example:Triangle BCA is right-angled at C if c = 27 and ∠A = 58°, find ∠B, b, and a.

B

A C

ac = 23

b = 17


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Solution:a. To find B, since B and ∠A are complementary angles, then

∠B + ∠ A = 90° ∠B = 90° – 58°

∠B = 32°

b. To find b, since b is the adjacent side of ∠A and c is the hypotenuse of right ∆ BCA, thenuse CAH.

cos θ =adjacent

hypotenuse

cos A =bc

cos 58° =b27

b = 27 cos 58°

b = 27 0.5299( )b = 14.31

c. To find a , since a is the opposite side of ∠A and c is the hypotenuse of right ∆ BCA, thenuse SOH.

sin θ =opposite

hypotenuse

sin A =ac

sin 58° =a27

a = 27 sin 58°

a = 27 0.8480( )a = 22.9

C. Solving a Right Triangle Given the Length of One Leg and the Measure of OneAcute Angle

Example:Triangle ACB is right-angled at C. If ∠A = 63° and a = 11 cm, find ∠B, b, and c .

B

C A b

a c = 27

58°


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Solution: a. To find ∠B, take note that ∠B and ∠A

are complementary angles. Then, ∠B + ∠ A = 90°

∠B = 90° – 63°

∠B = 27°

b. To find b, since b is the adjacent side c. To find c , since c is theand a is the opposite side of ∠A, then hypotenuse and a is the oppositeuse TOA. side of ∠A, then use SOH.

tan θ =oppositeadjacent

tan A =ab

tan 63° =11b

b tan 63° = 11

b 1.9626( )= 11

b =11

1.9626b = 5.60 cm

sin θ =opposite

hypotenuse

sin A =bc

sin 63° =11c

c sin 63° = 11

c 0.8910( )= 11

c =11

0.8910c = 12.35 cm

D. Solving a Right Triangle Given the Length of the Two Legs

Example:Triangle ACB is right-angled at C. If a = 18.5 cm and b = 14.2 cm, find c , ∠A, and ∠B.Solution:a. To find c , use Pythagorean theorem:

c 2

= a2

+ b2

c 2 = (18.5) 2 + (14.2) 2

c 2 = 342.25 + 201.64c 2 = 543.89

c = 543.89c = 23.32

A

C B

c b

a = 11

63°

A

C B

c b = 14.2 cm

a = 18.5


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b. To find ∠A, since a and b are c. Based on the fact that ∠A and ∠B areopposite and adjacent side of ∠A complementary, the measure of anglerespectively, then use TOA. ∠B is 90° – 52° = 38°

tan θ =oppos te

adjacent

an A =ab

an A =18.514.2

an A = 1.3028

We can use our scientific calculator to find an angle whose tangent is 1.3028.A = 52°

You have just learned the definition of the six trigonometric ratios. Make sure that you willbe able to use these in the succeeding activities.

What to PROCESS

In this stage, you will solve problems involving the measures of angles or sides of a righttriangle using the trigonometric ratios.

➤ Activity 3: Surfing the TrigonometryPlay the traditional Bingo Game as a class (activity with the teacher).1. Make your own BINGO card. Draw a 5 x 5 grid, and number the squares in random fashion

from 1 to 24 (assuming a “Free Space”) or 1 to 25.2. There must be a “Bingo caller” who will pick a random number from 1 to 24 by using a spin-

ner, picking numbers from a bag, or generating a random number from a graphing calculator.3. When he/she announces the number, your teacher will present a question. You solve the

problem and mark the box on your BINGO card that corresponds to the question number.4. BINGO is achieved in the normal manner – horizontally , diagonally, or vertically (or fourcorners, if you wish).

5. Raise your hands if you have BINGO.6. When somebody wins, everyone (including the winner) continues playing until all problems

have been solved, or until there are 5 or 6 winners.7. Your teacher will collect your card after the game.


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BINGO BOARD

Question:1. Did you enjoy the activity? Why?2. What have you learned from the activity?3. What mathematical concept did you apply to answer the problems?4. How will you use the concepts you have learned in your daily life?5. Is it fun to study math? Why?

Now, you have learned the definition of the six trigonometric ratios and you were able touse them in finding the missing sides or angles of an acute angle in the triangle. The nextactivity will help you master the concepts.


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Questions:1. How did you find your answers?2. What are the mathematical concepts that you have learned in the activity?3. Do you think you can apply these concepts in your daily life? How? Why?

You skills in finding the missing sides and angles of a right triangle were developed throughthe previous activities. The next one will help you reflect or understand and apply the sixtrigonometric ratios.

What to REFLECTand UNDERSTAND

In this section, your goal is to check your understanding on how the six trigonometric ratiosare used.

Try this out! Try this out!

Try this out!

Sketch a figure and solve each righttriangle ABC with right angle at C,given that:1. A = 15° and c = 372. B = 64° and c = 19.23. A = 15° and c = 254. A = 45° and c = 165. B = 56° and c = 16

Sketch a figure and solve each righttriangle ABC with right angle at C,given that:1. A = 76° and a = 132. A = 22° and b = 223. B = 30° and b = 114. B = 18° and a = 185. A = 77° and b = 42

Sketch a figure and solve each righttriangle ABC with right angle at C,given that:1. a = 15.8, b = 212. a = 7, b = 123. a = 2, b = 7

4. a = 3, b = 35. a = 250, b = 250


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➤ Activity 5: Use! List! Explain!Study the lesson. Then complete the following.1. Use the figure below to answer the following questions.

a. What is the length of side t ?b. What is the measure of ∠B?∠C?c. How did you find the measure of the angles?

2. List the trigonometric ratio or ratios of B that involve the measure of the hypotenuse.3. Explain how to determine which trigonometric ratio to use when solving for an unknown

part of a right triangle.

In the preceding activity, the discussion was about your knowledge on the six trigonometricratios. What are the new things that you have realized? Do you think you can use them in your daily life? How and why?

The next activity will help you answer these questions.

What to TRANSFER

In this section you will also learn how to make and use a clinometer, an improvisedinstrument used to measure the height of a certain object. Your goal now is to apply theknowledge that you have learned about the six trigonometric ratios.

➤ Activity 6: The ClinometerObjective: To make a clinometer and use it to measure the height of an object.Materials required: Stiff card, small pipe or drinking straw, thread, a weight

(a metal washer is ideal)Pre-requisite knowledge: Properties of right trianglesProcedure:(A) To make a clinometer:

1. Attach a protractor on a cardboard and fix a viewing tube (straw or pipe) along the diameter.2. Punch a hole (o) at the center of the semicircle.3. Suspend a weight {w} from a small nail fixed to the center.4. Ensure that the weight at the end of the string hangs below the protractor.5. Mark degrees (in sexagecimal scale with 00 at the lowest and 10 to 900 proceeding both

clockwise and counterclockwise). [Fig 1].

C

TB

t

15

8


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(B) To determine the height of an object:6. First measure the distance of the object from you. Let the distance be d .7. Look through the straw or pipe at the top of the object. Make sure you can clearly see the

top of the object.8. Hold the clinometer steady and let your partner record the angle the string makes on the

scale of the clinometer. Let this angle be θ .9. Determine the height of the object.10. Write your observation. You may change the distance of the object (by either moving the

object or by changing their position) and note how the angle of elevation varies.

Questions:1. What have you realized after doing the activity?

2. How did you find the height of the object?3. What other methods or devices can you suggest to solve the same type of problem?


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➤ Activity 7: Trigonome – Tree1. Look for a tree in your community then find

its height using a trigonometric ratio.2. Draw an illustration.3. Write a paragraph explaining how the

height of the tree was computed.4. Determine the other trigonometric ratios.5. What learning have you discovered in doing

such activity? Would you be able to use thisin your life? How and why?

In this section, your task was to measure the height of a tree using the trigonometric ratio.How did you find the performance task? How did the task help you realize the real-life appli-cation of the concepts?

SUMMARY/SYNTHESIS/GENERALIZATION

This lesson was about the six trigonometric ratios. Various activities were provided to help you illustrate and define the six trigonometric ratios. You also learned how to use themin finding the missing sides and angles of a right triangle and applied them to real-lifesituations. Your knowledge in this lesson will help you understand the next topic, which is,the trigonometric ratios of special angles.

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The Six Trigonometric Ratios