242 Trigonometric Functions

BIG IDEA Exact trigonometric values for multiples of 30º, 45º, and 60º can be found without a calculator from properties of special right triangles.

For most values of θ, the values of sin θ, cos θ, and tan θ cannot be found exactly and must be approximated. For this reason, you used approximate values found with a calculator in previous lessons.

In this lesson, you will apply what you know about 45º-45º-90º and 30º-60º-90º triangles to obtain exact values of cos θ, sin θ, and tan θ when θ is a multiple of 30º, 45º, or 60º.

Exact Values of Trigonometric Functions

for θ = 45º

You can use the properties of isosceles right triangles to fi nd cos 45º and sin 45º.

Example 1Use �OPF at the right to compute the exact values of cos 45º and sin

45º. Justify your answer.

Solution Because m∠FOP = 45º, m∠P = 45º. So �OPF is isosceles with legs −−

OF and ? . a and b are the lengths of the legs, so a = b. By the Pythagorean Theorem, a2 + b2 = 1, so 2a2 = 1, and a2 = ? . Therefore, a = b = ± 1 _

√ —

2 . Because a and b are lengths,

a = b = 1 _ √

— 2 .

But cos 45º = a and sin 45º = ? . Thus, cos 45º = sin 45º = 1 _

√ —

2 =

√ —

2 _ 2 .

QY1

Exact Values of Trigonometric Functions

for θ = 30º and θ = 60º

In Example 3 of Lesson 4-3, you were told that sin 30º = 1 _ 2 . You can verify this by using properties of equilateral triangles.

GUIDEDGUIDED

1

x

y

45˚a

b

O F

P = (a, b)= (cos 45˚, sin 45˚)1

x

y

45˚a

b

O F

P = (a, b)= (cos 45˚, sin 45˚)

QY1

Explain why tan 45º = 1.

QY1

Explain why tan 45º = 1.

Mental Math

A side of square SQUA, below, has length 5.

a. What is the length of

−−

AQ ?

b. If E is the midpoint of

−−

AQ , what is the length of

−−

SE ?

U

A S

Q

E

5

U

A S

Q

E

5

Lesson

Chapter 4

4-4Exact Values of Sines,

Cosines, and Tangents

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Exact Values of Sines, Cosines, and Tangents 243

Lesson 4-4

Example 2Derive the exact values of cos 30º and sin 30º.

Solution In equilateral �OPQ at the right, since OP = 1, PQ = ? . Consequently, PR = d = ? .

By the Pythagorean Theorem, c2 + d2 = 1. So c2 + ? = 1. c2 = ?

c = ?

Thus, (cos 30º, sin 30º) = (c, d) = ( ? , ? ).So, cos 30º = ? and sin 30º = ? .

To obtain the exact values of cos 60º and sin 60º, use the Complements Theorem: cos 60º = sin 30º = 1 _ 2 and sin 60º = cos 30º = √

__ 3 _ 2 .

Example 3Find the exact value of tan 30º.

Solution Use tan θ = sinθ

_ cosθ

.

tan 30º = sin 30º _ cos 30º = 1 _ 2 _

√ —

3 _ 2 = 1 _ 2 · 2 _

√ —

3 = 1 _

√ —

3 =

√ —

3 _ 3 .

QY2

You should memorize the exact values of cos θ, sin θ, and tan θ for θ = 30º, 45º, and 60º. They are important tools in mathematics and science because they are exact. To help you learn them, they are summarized below.

sin 45º = √ � 2

_ 2 = sin

π _

4 sin 30º = 1

_ 2 = sin

π _

6 sin 60º =

√ � 3

_ 2 = sin

π _

3

cos 45º = √ � 2

_ 2 = cos

π _

4 cos 30º =

√ � 3

_ 2 = cos

π _

6 cos 60º = 1

_ 2 = cos

π _

3

tan 45º = 1 = tan π

_ 4 tan 30º =

√ � 3

_ 3 = tan

π _

6 tan 60º = √ � 3 = tan

π _

3

QY3

GUIDEDGUIDED

1

Q

O

P = (c, d)= (cos 30º, sin 30º)

x

y

60˚30˚

c

d

R

1

Q

O

P = (c, d)= (cos 30º, sin 30º)

x

y

60˚30˚

c

d

R

QY2

Find the exact value of tan 60º.

QY2

Find the exact value of tan 60º.

45˚

1

x

y

22

22

22

22

,

1x

y

32

32 ,

30˚

12

12

1

x

y

60˚

32

32

,12

12

45˚

1

x

y

22

22

22

22

,

1x

y

32

32 ,

30˚

12

12

1

x

y

60˚

32

32

,12

12

QY3

Which theorem verifi es that sin 30º = cos 60º?

QY3

Which theorem verifi es that sin 30º = cos 60º?

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244 Trigonometric Functions

Chapter 4

Exact Values for Sines and Cosines of Multiples

of 30º, 45º, and 60º

Using the defi nitions of sine and cosine and the Symmetry Identities, you can fi nd exact values of the trigonometric functions for all integer multiples of 30º, 45º, and 60º.

Example 4Find exact values of cos 120º, sin 120º, and tan 120º.

Solution By the Supplements Theorem,

cos 120º = ? = – 1 _ 2 and sin 120º = ? = √ � 3 _ 2 .

tan 120º = sin 120º _ cos 120º = √ � 3

_ 2 _

– 1 _ 2 = –√

— 3 .

Check Use a calculator.

On the unit circle below are the images of (1, 0) under rotations of integer multiples of 30º or 45º between 0º and 360º. You should be able to calculate exact values of the sine, cosine, and tangent functions for all pictured values of θ by relating them to one of the points in the fi rst quadrant or on the axes.

Copy the unit circle and the exact values

of (cos θ, sin θ) given at the right. Use

your knowledge of refl ections and

symmetries to add the exact values

of trigonometric functions for multiples of

30º, 45º and 60º in Quadrants II, III,

and IV.

Exact Values for Trigonometric Functions of Radians

It is important to know the exact values of trigonometric functions for certain radians. You can compute those values by converting to degrees, but in the long run, it is helpful to learn to “think radian.”

GUIDEDGUIDED

y

x60˚60˚

(cos 60˚, sin 60˚)(cos 120˚, sin 120˚)y

x60˚60˚

(cos 60˚, sin 60˚)(cos 120˚, sin 120˚)

= (cos 120˚, sin 120˚)(cos 135˚, sin 135˚)

(cos 150˚, sin 150˚)(-1, 0) = (cos 180˚, sin 180˚)

(cos 210˚, sin 210˚)(cos 225˚, sin 225˚)

(cos 240˚, sin 240˚)

(cos 30˚, sin 30˚) =

(cos 45˚, sin 45˚) =

(cos 60˚, sin 60˚) =

(cos 270˚, sin 270˚) = (0, -1)

(cos 0˚, sin 0˚) = (1, 0)

(cos 300˚, sin 300˚)(cos 315˚, sin 315˚)

(cos 330˚, sin 330˚)

y

x0

32

12- ,

22

22,

,3

212

32 ,

12

= (cos 120˚, sin 120˚)(cos 135˚, sin 135˚)

(cos 150˚, sin 150˚)(-1, 0) = (cos 180˚, sin 180˚)

(cos 210˚, sin 210˚)(cos 225˚, sin 225˚)

(cos 240˚, sin 240˚)

(cos 30˚, sin 30˚) =

(cos 45˚, sin 45˚) =

(cos 60˚, sin 60˚) =

(cos 270˚, sin 270˚) = (0, -1)

(cos 0˚, sin 0˚) = (1, 0)

(cos 300˚, sin 300˚)(cos 315˚, sin 315˚)

(cos 330˚, sin 330˚)

y

x0

32

12- ,

22

22,

,3

212

32 ,

12

ActivityActivity

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Exact Values of Sines, Cosines, and Tangents 245

Lesson 4-4

Example 5Without using technology, compute the exact value of each trigonometric

function below.

a. sin π

_ 4 b. cos

5π _

6 c. tan π

Solution

a. Convert to degrees: π

_ 4 · 180º _

π = 45º. sin 45º = √

— 2 _ 2 = sin π _ 4

b. 5π

_ 6 = ? º, so cos 5π _ 6 = ? .

c. tan π = sin π

_ cos π = ? _ ? = ?

Questions

COVERING THE IDEAS

In 1–3, refer to the unit circle at the right in which m∠POA = 30º,

m∠QOA = 45º, and m∠ROA = 60º. Name a segment whose length

equals the following.

1. cos 30º 2. sin 45º 3. sin 60º

4. Evaluate.

a. cos π

_ 3 b. tan π

_ 4 c. sin π

_ 6

In 5–10, fi nd the exact value.

5. a. sin 240º b. cos 240º c. tan 4π

_ 3

6. a. sin 3π

_ 4 b. cos 3π

_ 4 c. tan 135º

7. a. sin 11π

_ 6 b. cos(–30º) c. tan 11π

_ 6

8. sin 210º 9. cos 5π

_ 3 10. tan(–405º)

11. Draw a unit circle as in the Activity, labeling the angles in radians

and fi lling in all the values of the trigonometric functions.

APPLYING THE MATHEMATICS

12. a. Find two values of θ between –90º and 90º for which cos θ = 1 _ 2 .

b. Find two values of θ between 270º and 450º for which cos θ = 1 _ 2 .

c. What is the relationship between the two pairs of angles formed

in Parts a and b?

13. Consider the equation sin θ = –

1 _

2 .

a. Draw a unit circle and mark the two points for which sin θ = –

1 _

2 .

b. Give two values of θ between 0º and 360º that satisfy

the equation.

c. Give two values of θ between 0 and 2π radians that satisfy

the equation.

14. a. Find two values of θ between 0 and 2π such that cos θ = sin θ.

b. What is the value of tan θ for each value of θ in Part a?

GUIDEDGUIDED

O AH G F

R QP

x

y

O AH G F

R QP

x

y

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246 Trigonometric Functions

Chapter 4

15. True or False If tan θ = ±1, then θ = (45n)º and n is an odd integer. Justify your answer.

16. The regular nonagon ABCDEFGHI pictured here is inscribed in the unit circle.

a. Give the exact coordinates of point B in terms of θ. b. Give the value of θ in radians. c. Estimate AB to the nearest thousandth.

REVIEW

17. Without using a calculator, given that sin 52º ≈ 0.788, estimate each value. (Lesson 4-3)

a. sin(–52º) b. sin 128º c. sin 232º d. cos 38º

18. True or False For all θ, cos(θ + 90º) = sin θ. (Lesson 4-3)

19. Without using a calculator, give the exact value for sin ( – π

_ 2 ) . (Lesson 4-2)

20. a. Prove that cos θ · tan θ = sin θ for all cos θ ≠ 0. b. Why is it impossible to have cos θ = 0 in Part a? (Lesson 4-2)

21. Convert the following measures to radians. (Lesson 4-1)

a. 135º b. 390º c. –215º d. –270º

In 22 and 23, consider g(t) = t2 + 1 and f(t) = 3t – 1. (Lesson 3-7)

22. Evaluate g(f(–80)). 23. Find a formula for (f ◦ g)(t).

24. When a certain drug enters the blood stream, its potency decreases exponentially with a half-life of 8 hours. Suppose the initial amount of drug present is A. How much of the drug will be present after each number of hours? (Lesson 2-5)

a. 8 b. 24 c. t

EXPLORATION

25. A regular triangle, hexagon, and dodecagon have been inscribed in the unit circle. Find the exact perimeter of each polygon. You may fi nd a CAS useful.

x

y

A

B

CD

E

F

G H

I

θ x

y

A

B

CD

E

F

G H

I

θ

x

y

x

y

x

y

x

y

x

y

x

yQY ANSWERS

1. tan 45º = sin 45º _ cos 45º

= √

__ 2 _ 2 _

√

__ 2 _ 2 = 1

2. tan 60º = sin 60º _ cos 60º = √

3 _ 2 _

1 _ 2

= √__

3

3. Complements Theorem

QY ANSWERS

1. tan 45º = sin 45º _ cos 45º

= √

__ 2 _ 2 _

√

__ 2 _ 2 = 1

2. tan 60º = sin 60º _ cos 60º = √

3 _ 2 _

1 _ 2

= √__

3

3. Complements Theorem

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