Chapter 4
Lesson
Exact Values of Sines,
Cosines, and Tangents
4-4
BIG IDEA
Exact trigonometric values for multiples of 30º,
45º, and 60º can be found without a calculator from properties of
special right triangles.
Mental Math
A side of square SQUA,
below, has length 5.
U
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.
Q
E
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.
A
5
S
a. What is the length
−−
of AQ?
Exact Values of Trigonometric Functions
for θ = 45º
b. If E is the midpoint
−−
of AQ, what is the length
−−
of SE?
You can use the properties of isosceles right triangles to find cos 45º
and sin 45º.
GUIDED
Example 1
y
Use 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 = ? .
1
Therefore, a = b = ± _
— . Because a and b are lengths,
√2
1
a = b = _.
1
O
45˚
a
P = (a, b)
= (cos 45˚, sin 45˚)
b
x
F
—
√2
But cos 45º = a and sin 45º = ? .
—
√2
1
Thus, cos 45º = sin 45º = _ = _.
—
√2
2
QY1
QY1
Explain why tan 45º = 1.
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 . You can
2
verify this by using properties of equilateral triangles.
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Trigonometric Functions
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Lesson 4-4
GUIDED
Example 2
y
Derive the exact values of cos 30º and sin 30º.
Solution In equilateral OPQ at the right, since OP = 1,
PQ = ? . Consequently, PR = d = ? .
1
By the Pythagorean Theorem, c2 + d2 = 1.
So c2 + ? = 1.
O
c2 =
?
c =
?
30˚
60˚
P = (c, d)
= (cos 30º, sin 30º)
d
c
x
R
Q
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
__
√3
Theorem: cos 60º = sin 30º = _1 and sin 60º = cos 30º = _
.
2
2
Example 3
Find the exact value of tan 30º.
sinθ
.
Solution Use tan θ = _
cosθ
_1
—
√3
sin 30º
2
2
1
_1 · _
_
_
tan 30º = _ = _
— =
— = — = 3 .
√3
2 √3
√3
cos 30º
_
2
QY2
QY2
Find the exact value of
tan 60º.
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.
√
2
π
_
sin 45º = _
2 = sin 4
√
2
π
_
cos 45º = _
2 = cos 4
π
1
_
sin 30º = _
2 = sin 6
√
3
π
_
cos 30º = _
2 = cos 6
√
3
π
_
sin 60º = _
2 = sin 3
π
1
cos 60º = _ = cos _
π
tan 45º = 1 = tan _
4
√
3
π
_
tan 30º = _
3 = tan 6
π
tan 60º = √
3 = tan _
3
y
y
y
2 2
2 , 2
1
45˚
2
2
2
2
x
1
30˚
3
2
3 1
2 ,2
1
2
x
2
1
60˚
1
2
3
1 3
2,2
3
2
x
QY3
QY3
Which theorem verifies
that sin 30º = cos 60º?
Exact Values of Sines, Cosines, and Tangents
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Chapter 4
Exact Values for Sines and Cosines of Multiples
of 30º, 45º, and 60º
Using the definitions of sine and cosine and the Symmetry Identities,
you can find exact values of the trigonometric functions for all integer
multiples of 30º, 45º, and 60º.
GUIDED
Example 4
Find exact values of cos 120º, sin 120º, and tan 120º.
Solution By the Supplements Theorem,
cos 120º = ? = – _1 and sin 120º =
2 √
3
_
?
(cos 120˚, sin 120˚)
√
3
=_
.
2
y
(cos 60˚, sin 60˚)
x
60˚ 60˚
—
sin 120º
2
tan 120º = _ = _
= –√ 3 .
cos 120º
– _1
Check Use a calculator.
2
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 first
quadrant or on the axes.
Activity
Copy the unit circle and the exact values
of (cos θ, sin θ) given at the right. Use
your knowledge of reflections and
symmetries to add the exact values
of trigonometric functions for multiples of
30º, 45º and 60º in Quadrants II, III,
and IV.
y
- 12
,
3
2
(cos 60˚, sin 60˚) =
= (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˚)
1
2
3
, 2
(cos 45˚, sin 45˚) =
0
2
2
,
3
2
2
2
1
(cos 30˚, sin 30˚) =
,2
x
(cos 0˚, sin 0˚) = (1, 0)
(cos 330˚, sin 330˚)
(cos 315˚, sin 315˚)
(cos 300˚, sin 300˚)
(cos 270˚, sin 270˚) = (0, -1)
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.”
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Trigonometric Functions
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Lesson 4-4
GUIDED
Example 5
Without using technology, compute the exact value of each trigonometric
function below.
5π
π
b. cos _
c. tan π
a. sin _
4
6
Solution
—
√2
180º
π _
π
_
_
a. Convert to degrees: _
4 · π = 45º. sin 45º = 2 = sin 4
5π
5π
b. _ = ? º, so cos _ = ? .
6
6
sin π
_?
c. tan 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
y
R
3
O
π
b. tan _
4
In 5–10, find the exact value.
5. a. sin 240º
b. cos 240º
3π
3π
_
6. a. sin 4
b. cos _
4
11π
7. a. sin _
b. cos(–30º)
6
8. sin 210º
5π
9. cos _
3
P
3. sin 60º
4. Evaluate.
π
a. cos _
Q
H G F A
x
π
c. sin _
6
4π
c. tan _
3
c. tan 135º
11π
c. tan _
6
10. tan(–405º)
11. Draw a unit circle as in the Activity, labeling the angles in radians
and filling 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 = _12 .
b. Find two values of θ between 270º and 450º for which cos θ = _1 .
2
c. What is the relation between the two pairs of angles formed in
Parts a and b?
13. Consider the equation sin θ = – _12 .
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?
Exact Values of Sines, Cosines, and Tangents
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Chapter 4
15. True or False If tan θ = ±1, then θ = (45n)º and n is an odd
integer. Justify your answer.
y
C
D
16. The regular nonagon ABCDEFGHI pictured here is inscribed in
B
E
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.
θ
A
x
F
I
G
H
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)
π
19. Without using a calculator, give the exact value for sin(– _
.
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) = t 2 + 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
find a CAS useful.
y
y
y
QY ANSWERS
sin 45º
1. tan 45º = _
cos 45º
x
x
x
__
√2
_
2__
=_
=1
√
2
_
2
3
_
√2
sin 60º _
2. tan 60º = _
= _1
cos 60º
__
2
= √3
3. Complements Theorem
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