Module - Trigonometric Functions of an Acute Angletlment.nait.ca/.../T06-Trigonometric_Functions_of_Acute_Angles.pdf · Trigonometric Functions of an Acute Angle. ... The three sides

LAST REVISED December, 2008

Copyright This publication © The Northern Alberta Institute of Technology 2002. All Rights Reserved.

Trigonometry Module T06

Trigonometric Functions of an Acute Angle

Module T06 − Trigonometric Functions of Acute Angles

Trigonometric Functions of an Acute Angle Statement of Prerequisite Skills Complete all previous TLM modules before completing this module.

Required Supporting Materials Access to the World Wide Web. Internet Explorer 5.5 or greater. Macromedia Flash Player.

Rationale Why is it important for you to learn this material? Trigonometry is simply the study of triangles. It has been used since the time of the Greeks. The trigonometric ratios introduced in this module are used in a wide variety of applications today including surveying, navigation, engineering, and construction.

Learning Outcome When you complete this module you will be able to… Solve problems involving the six trigonometric functions.

Learning Objectives 1. Define and list the six trigonometric functions in terms of the sides of a right triangle. 2. Determine the function values of angles between 0 and 90 degrees inclusively. 3. Determine the angle from a given function value. 4. Calculate the function value of the acute angles in any right triangle, given two sides

of the triangle. 5. Determine all function values of an acute angle given one function value of the angle.

Connection Activity Consider the following Triangle:

B C

A

b c = 5

a = 4 Your work with right triangles allows you to solve for side b. What is the ratio of side a to side c? What if a = 8 and c = 10? What is the ratio of side a to side c now? Notice the ratio has not changed. The proportionate increase in the sides a and c means that angle B did not change and the ratio of the sides remained the same. This is a hint at how the ratio of two sides of a triangle can help you determine the angle between the sides.

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http://www.microsoft.com/windows/ie/default.asp

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OBJECTIVE ONE When you complete this objective you will be able to… Define and list the six trigonometric functions in terms of the sides of a right triangle.

Exploration Activity Define and list the six trigonometric functions in terms of the sides of a right triangle. There are six possible trigonometric functions. Following is a list of these functions and their abbreviations.

The basic functions: sine - sin cosine - cos tangent - tan

The reciprocals of the basic functions:

cosecant - csc secant - sec cotangent - cot

The above trigonometric functions are meaningless in themselves, you must relate them to angles of a triangle. These side-angle relationships are illustrated in the following right angle triangle that identifies the sides relative to θ. These relationships hold true only for triangles with a 90º angle. These relationships must be memorized:

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sin θ = hypotenuse

opposite

Opposite θ

Adjacent to θ

Hypotenuse

θ

opposite

cos θ = hypotenuse

adjacent

tan θ = adjacentopposite

csc θ = opposite

hypotenuse = θsin

1

sec θ = adjacent

hypotenuse = θcos

1

cot θ = oppositeadjacent =

θtan1


NOTE: 1. The sin and csc functions are related in that the csc is the reciprocal of the sin. 2. The cos and sec functions are reciprocals of each other. 3. The tan and cot functions are also reciprocals of each other. We can name our right triangle ABC, and identify the functions in terms of the lettered sides.

sin A = ca

A C

B

ac

b

cos A = cb

tan A = ba

csc A = ac

sec A = bc

cot A = ab

NOTE: When labeling a triangle the same letter is used for the angle and the side opposite it. Use capitals for angles and lowercase for sides. Observe the reciprocal relationships in these fractions, i.e.

1sincsc

aAc A

= =

csc A = ca

1cos

secbAc A

= =

sec A = cb

1tan

cotaAb A

= = cot A = ba

3


If we wished to find the 6 functions of angle B in the previous diagram we would obtain.

sin B = cb

hypopp

= csc B = bc

opphyp

=

cos B = ca

hypadj

= sec B = ac

adjhyp

=

tan B = ab

adjopp

= cot B = ba

oppadj

=

Observe again the reciprocal relationships in the above functions. Sometimes students may get confused as to which side is the "opposite" side from an angle. Remember, the "opposite" side is the side which is not an arm of the angle. The hypotenuse is always opposite the right angle. The adjacent side is always an arm of the angle.

EXAMPLE 1 For the triangle shown, determine the ratios of the six trigonometric functions of angle θ.

EXACT DECIMAL

12

16

20

θ

222 121620 +=

sin θ = 2012

53 0.6000

cos θ = 2016

54 0.8000

tan θ = 1612

43 0.7500

csc θ = 1220

35 1.6667

sec θ = 1620

45 1.2500

cot θ = 1216

3Pythagorean Theorem Check 4 1.3333

4


csc, sec, cot could have also been found by using the reciprocals of the first 3 trig functions as shown in the next 3 examples. 1. sin θ = 0.60000

1 1csc 1.6667sin 0.6000

θθ

= = =

2. cos θ = 0.8000

secθ = 1 1 1.2500cos 0.8000θ

= =

3. tan θ = 0.7500

1 1cot 1.3333tan 0.7500

θθ

= = =

5


Experiential Activity One 1. In the figure, choose either angle P or angle R.

a) OROP = tan ? P

O

R

b) OPPR = sec ?

c) PROP = cos ?

d) PROP = sin ?

e) OPPR = csc ?

2. The three sides of a right triangle are 5, 12, and 13. Let A be the acute angle opposite the side 5 and let B be the other acute angle. Find the six functions of A and B. Show Me

3. In the figure, if x = r, find the six functions of θ. R

z x

r θ

ZX

4. In the figure in question 3, x = 2r, find the sine, cosine, and tangent of θ.

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Experiential Activity One Answers 1. a) R, b) P, c) P, d) R, e) R 2. sin A = 0.3846 sin B = 0.9231 cos A = 0.9231 cos B = 0.3846 tan A = 0.4167 tan B = 2.4000 csc A = 2.6000 csc B =1.0833 sec A =1.0833 sec B = 2.6000 cot A = 2.4000 cot B = 0.4167

3. sin ; cos ; tan 1; csc ; sec ; cos 1x r z zz z x r

θ θ θ θ θ θ= = = = = =

4. 2sin ; cos ; tan 2r rz z

θ θ θ= = =

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OBJECTIVE TWO When you complete this objective you will be able to… Determine the function values of angles between 0 and 90 degrees inclusively.

Exploration Activity Now we will use the calculator to find the ratios of the 6 trigonometric functions. Refer to your calculator manual for this objective. To find the SINE, COSINE, and TANGENT of any angle, you enter the desired function and the desired angle. EXAMPLE Take: 32° a) Find the trigonometric ratio of sin 32º

Step 1: press the sin key Step 2: press 32º and then the = key Step 3: display: 0.5299

b) cos 32° Step 1: press the cos key Step 2: enter 32º Step 3: display: 0.8480

c) tan 32°

Step 1: press the tan key Step 2: enter 32º Step 3: display: 0.6249

Ensure calculator is in degree mode when the angle is given in degrees and you are using any of the trig functions on your calculator!!

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EXAMPLES TRIGONOMETRIC RATIOS OF RECIPROCAL FUNCTIONS Finding the COSECANT, SECANT, and COTANGENT of any angle is explained in the following examples:

Find csc 79º, sec 79º and cot 79º to four decimal places: PREFERRED METHOD ALTERNATE METHOD

Example 1:

csc 79º

Find the sin 79º, then take the reciprocal. This will be equal to the csc 79º. Step 1: press the sin key. Step 2: press 79 and then the = key. Step 3: press the x −1 key. Step 4: display: 1.0187 (= csc 79º)

Since csc 79º = 1sin 79°

Enter: 1 ÷ sin 79 = Display: 1.0187

Example 2:

sec 79º

Find the cos 79º, then take the reciprocal. This will be equal to the sec 79º. Step 1: press the cos key. Step 2: press 79 and then the = key. Step 3: press the x −1 key. Step 4: display: 5.2408 (= sec 79º)

Since sec 79º = 1cos79°

Enter: 1 ÷ cos 79 = Display: 5.2408

Example 3:

cot 79º

Find the tan 79º, then take the reciprocal. This will be equal to the cot 79º. Step 1: press the tan key. Step 2: press 79 and then the = key. Step 3: press the x −1 key. Step 4: display: 0.1944 (= cot 79º)

Since cot 79º = 1tan 79°

Enter: 1 ÷ tan 79 = Display: 0.1944

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Module T06 − Trigonometric Functions of Acute Angles 10

Experiential Activity Two Exercise Set A Perform the following: 1. sin 64° 2. cos 75° 3. tan 74° 4. tan 12º 5. cos 15º 6. sin 17° 7. cos 28° 8. tan 34° 9. cos 63° 10. sin 30° Exercise Set B Perform the following: 1. csc 43° 2. cot 14° 3. tan 55° 4. sec 35° 5. csc 30° 6. cos 55° 7. cot 15° 8. sec 14° Show Me 9. csc 4° 10. sin 18° 11. sin 0° 12. tan 0°

Experiential Activity Two Answers Answers for Exercise Set A 1. 0.8988 2. 0.2588 3. 3.4874 4. 0.2126 5. 0.9659 6. 0.2924 7. 0.8829 8. 0.6745 9. 0.4540 10. 0.5000 Answers for Exercise Set B 1. 1.4663 2. 4.0108 3. 1.4281 4. 1.2208 5. 2.0000 6. 0.5736 7. 3.7321 8. 1.0306 9. 14.3356 10. 0.3090 11. 0.0000 12. 0.0000


OBJECTIVE THREE When you complete this objective you will be able to… Determine the angle from a given function value.

Exploration Activity

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The trigonometric ratio of 0.5976 is the ratio of the sides hypotenuse

opposite for θ = 36.7º.

When we are given sin θ = 0.5976 and are asked to determine θ, we know that the only acute angle which has a sine of 0.5976 is 36.7º. Therefore, we should be able to find the angle.

Example:

sin θ = 0.5976 θ = arc sin 0.5976 θ = 36.7º

or sin θ = 0.5976 θ = sin−1 0.5976 θ = 36.7º

Arc sin and sin−1 are the two names used to express this operation.

It is very important you understand that sin-1 θ means the inverse operation of finding the

sine of the angle.

sin−1 θ is not the same as 1sinθ

Do not confuse this with the reciprocal x−1 function on your calculators!!

EXAMPLE 1 Given: sin θ = 0.5976. Find θ in degrees. SOLUTION: The problem is to find the angle. Therefore by our definition θ = Arc sin 0.5976 or θ = sin−1 0.5976 This means we want to find the angle whose sine is 0.5976


Using the calculator: Refer to your calculator manual as the procedure varies from calculator to calculator. Step 1: Press the 2nd F key Step 2: press sin key Step 3: enter 0.5976 and then press the = key Step 4: display: 36.7º Check: sin 36.7º = 0.5976 NOTE: θ = Arc sin 0.5976 is read as "θ is the angle whose sine is 0.5976". θ = Arc cos 0.3276 is read as "θ is the angle whose cosine is 0.3276 . θ = Arc tan 4.364 is read as "θ is the angle whose tangent is 4.364" .... and so on. It is sometimes desirable to find the angle in radians. Again your calculator will do this operation as long as you change it to radian mode.

EXAMPLE 2 Find the value of θ in radians for cos θ = 0.9690 θ = Arc cos 0.9690 SOLUTION: Again refer to your own calculator manual for finding an angle in radians, given a trigonometric function. Using your calculator: Step 1: Put your calculator in radian mode Step 2: Press the 2nd F key Step 3: press cos key Step 4: enter 0.9690 and then press the = key Step 5: display: 0.2496 Therefore θ = 0.2496 Check: cos(0.2496) = 0.9690 Using only csc, sec and cot, find the angle in degrees or radians.


EXAMPLE 3 Given sec θ = 5.2411, find angle θ in degrees Since the calculators do not have a secant function, it is necessary first to convert this to the cosine function. Since

cos θ = 1secθ

cos θ = 15.2411

θ = arc cos 15.2411

⎛ ⎞⎜ ⎟⎝ ⎠

or

θ = cos−1 15.2411

⎛ ⎞⎜ ⎟⎝ ⎠

θ = 79.0º

EXAMPLE 4 Given csc θ = 1.7965, find angle θ in degrees SOLUTION: Since

sin θ = 1cscθ

sin θ = 11.7965

θ = arc sin 1⎝ ⎠1.7965⎛ ⎞⎜ ⎟ or

θ = sin-1 1⎛ ⎞1.7965⎜ ⎟

⎝ ⎠

θ = 33.8º

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Experiential Activity Three Exercise Set A Solve for θ in degrees, given: 1. sin θ = 0.9135 2. cos θ = 0.8241 3. tan θ = 0.4142 4. sin θ = 0.4305 5. tan θ = 1.2059 6. cos θ = 1.2000 7. sin θ = 0.7071 8. tan θ = 0.2493 9. cos θ = −1.0000 Exercise Set B Find θ in Radians, given: 1. sin θ = 0.5821 2. cos θ = 0.4555 3. tan θ = 3.5105 4. sin θ = 0.3778 5. tan θ =1.0761 6. cos θ = 0.0279 7. sin θ = 0.7804 8. cos θ = 0.9763 Exercise Set C (a) Find the angle in degrees, given the following: 1. csc θ =1.970 2. sec θ = 2.000 3. cot θ =1.962 4. csc θ = 5.495 5. sec θ = 6.795 6. cot θ = 6.543 7. sec θ = 3.540 8. cot θ = 6.354 9. csc θ = 4.945 Show Me 10. sec θ = 6.514 (b) For the exercise above, find the angles in radians.


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Experiential Activity Three Answers Answers for Exercise Set A 1. 66.0º 2. 34.5º 3. 22.5º 4. 25.5º 5. 50.3º 6. none this is because cos θ and sin θ are limited to values between −1 and 1 7. 45.0º 8. 14.0º 9. 180º Answers for Exercise Set B 1. 0.6213 2. 1.0979 3. 1.2933 4. 0.3874 5. 0.8220 6. 1.5429 7. 0.8953 8. 0.2181 Answers for Exercise Set C (a.) (b.) 1. 30.5º 1. 0.5324 2. 60.0º 2. 1.0472 3. 27.0º 3. 0.4714 4. 10.5º 4. 0.1830 5. 81.5º 5. 1.4231 6. 8.7º 6. 0.1517 7. 73.6º 7. 1.2844 8. 8.9º 8. 0.1561 9. 11.7º 9. 0.2036 10. 81.2º 10. 1.4167


OBJECTIVE FOUR When you complete this objective you will be able to… Calculate the function value of the acute angles in any right triangle, given two sides of the triangle.

Exploration Activity If we are given any two sides of a right triangle then according to Pythagoras Theorem we can calculate the third side. This theorem states that in a right triangle the square of the hypotenuse equals the sum of the squares on the other two sides. In the adjoining triangle we can find side b by using Pythagoras' formula: c2 = a2 + b2

52 = 42 + b2 b2 = 25 − 16

9=b

so: b = 3

B C

A

bc = 5

a = 4

Once we know the three sides, we can easily find the trigonometric ratios of both angle A and angle B

Trigonometric Ratios of ∠A

16

B C

b =3 c = 5

a = 4 Opposite ∠A

Adjacent to

A

∠A

hypotenuse

Trigonometric Ratios of ∠B

B C

b =3 c = 5

a = 4 Adjacent to

A

∠B

Opposite ∠B

hypotenuse

sin A = hypopp =

54 = 0.8000 sin B =

hypopp =

53 = 0.6000

cos A = 53 = 0.6000 cos B =

54 = 0.8000

tan A = 34 = 1.3333 tan B =

43 = 0.7500

csc A = 45 = 1.2500 csc B =

35 = 1.6667

sec A = 35 = 1.6667 sec B =

45 = 1.2500

cot A = 43 = 0.7500 cot B =

34 = 1.3333


EXAMPLE 1 If a = 3 and b = 4 in the right triangle ABC to the right, find sin A and tan B. since c2 = a2 + b2

c2 = 42 + 32 c2 = 25

25=c 222 bac +=

c = 5 so:

A C

a = 3 c

B

b = 4

sin A = 53 = 0.6000 and

tan B = 34 = 1.3333

NOTE: The trigonometric ratio is equally acceptable in either fraction form or in decimal form. See example 1 above. However when entering answers into the computer the decimal form must be used. i.e. 3/5 would be entered as 0.6000. Enter answers with 4 decimal places.

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Experiential Activity Four 1. Find the indicated ratios from the given triangle where a = 1, b = 3 , and c = 2.

Write your ratios in exact form.

a) sin A, sec B, cot A b) csc A, sin B, cot B c) tan A, cos B, sec A d) tan B, csc B, cos A

A C

B

a c

b

2. Find the indicated ratios from the given triangle where b =2.89, and c = 3.41, and side a is unknown. Round your ratios to 3 significant digits.

a) sin A, sec B, cot A b) csc A, sin B, cot B c) tan A, cos B, sec A d) tan B, csc B, cos A

A C

B

a c

b 3. Determine the indicated ratios. The listed sides are those shown in the given triangle.

Write your ratios in exact form.

a) a = 3, b = 4. Find tan A and cos B b) a = 5, c = 13. Find cos A and csc B c) b = 9, c = 41. Find cot A and cos B d) a = 8, c = 19. Find sin A and sec B

Round your ratios to 3 significant digits.

e) a = 2, c = 4. Find sec A and tan B f) b = 14, c = 23. Find csc A and cos B g) a = 132, b = 75. Find sin A and tan B

A C

B

ac

b

18


4. Given the coordinates as shown, find the values requested:

Round trigonometric ratios to 3 significant digits. Round angles to the one decimal place.

tan A = __________ ∠ A = __________ csc A = __________ Show Me

Y

X

A

P(3.6, 4.2)

5. Given the coordinates as shown, find the values requested: Round trigonometric ratios to 3 significant digits. Round angles to the one decimal place. ∠ B = __________ sin B = __________ sec B = __________

Y

X B

P(5.8, 2.3)

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Experiential Activity Four Answers

1. a) sin A = 21 , sec B = 2, cot A = 3

b) csc A = 2, sin B = 23 , cot B =

31

c) tan A = 3

1 , cos B = 21 , sec A =

32

d) tan B = 3 , csc B = 3

2 , cos A = 23

2. a) sin A = 0.531, sec B = 1.88, cot A = 1.60 b) csc A = 1.88, sin B = 0.848, cot B = 0.626 c) tan A = 0.626, cos B = 0.531, sec A = 1.18 d) tan B = 1.60, csc B = 1.18, cos A = 0.848

3. a) tan A = 43 , cos B =

53

b) cos A = 1312 , csc B =

1213

c) cot A = 409 , cos B =

4140

d) sin A = 198 , sec B =

819

e) sec A = 1.15, tan B = 1.73 f) csc A = 1.26, cos B = 0.793 g) sin A = 0.869, tan B = 0.568 4. a) tan A = 1.17, ∠A = 49.4º, csc A = 1.32 5. a) ∠B = 21.6º, sin B = 0.369, sec B = 1.08

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OBJECTIVE FIVE When you complete this objective you will be able to… Determine all function values of an acute angle given one function value of the angle.

Exploration Activity

EXAMPLE 1 Find the values of the trigonometric ratios of ∠A B

A C

a c

b

given that sin A = 1312 .

We know that sin A = hypotenuse

opposite

Therefore:

sin A = 1312

Since 12 is the side opposite ∠A and 13 is the hypotenuse we can identify the following sides: a = 12 and c = 13 Use the Pythagorean theorem to find side b: c2 = a2 + b2

132 = 122 + b2 25=b

B

b = 5

a =12

b = 5 Adjacent to ∠ A

Opposite ∠ A

Hypotenuse

A C

c = 13

Trigonometric ratios of ∠A

21

sin A = 1312 = 0.9231 cos A =

135 = 0.3846 tan A =

512 = 2.4000

csc A = 1213 = 1.0833 sec A =

513 = 2.6000 cot A =

125 = 0.4167


EXAMPLE 2

Given that tan A = 41 , find sin A and csc A.

Since tan A = adjopp and from the question tan A =

41 , we can identify the following sides

on the triangle below: a = 1 and b = 4

A C

B

a c

b

Use the Pythagorean Theorem to find side c: c2 = a2 + b2

c2 = 12 + 42 17=c

sin A = hypopp = 1

17 = 0.2425

csc A = opphyp = 17

1 = 4.1231

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Experiential Activity Five 1. Find the values of the other five trigonometric functions of A, given in exact values.

Write your answers in exact form.

a) sin A = 257 b) cot A =

724 c) tan A =

53

d) cos A = 21 e) cos A =

117 f) cot A =

23

Experiential Activity Five Answers

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1. a) cos A = 2524 , tan A =

247 , csc A =

725 , sec A =

2425 , cot A =

724

b) sin A = 257 , cos A =

2524 , tan A =

247 , csc A =

725 , sec A =

2425

c) sin A = 343 , cos A =

345 , csc A =

334 , sec A =

534 , cot A =

35

d) sin A = 23 , tan A = 3 , csc A =

32 , sec A =2, cot A =

3 1

e) sin A = 1172 , tan A =

772 , csc A =

7211 , sec A =

711 , cot A =

727

f) sin A = 7

2 , cos A = 73 , tan A =

32 , csc A =

27 , sec A =

37

Practical Application Activity Complete the Trigonometric Functions of an Acute Angle assignment in TLM.

Summary This module introduced the student to the six trigonometric functions.

Documents

Module - Trigonometric Functions of an Acute Angletlment.nait.ca/.../T06-Trigonometric_Functions_of_Acute_Angles.pdf · Trigonometric Functions of an Acute Angle. ... The three sides