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Trigonometric
Identities a n d
Conditional
Equations
Outline
Application
6-1 Basic Identities and Their Use
The roof for a 24-foot-wide industrial building is
formed by bending a 30-foot-wide panel of
corrugated steel into a circular arc (see the figure).
How high (to the nearest tenth of a foot) is the
building?
6-2 Sum, Difference, and
Cofunction Identities
6-3 Double-Angle and Half-Angle
Identities
–Sum and
6-4 Product–
–Product Identities
Sum–
30 ft
6-5 Trigonometric Equations
Chapter 6 Group Activity: From
M sin Bt ⴙ N cos Bt to
—A Harmonic
A sin (Bt ⴙ C)—
Analysis Tool
Chapter 6 Review
10 ft
24 ft
rigonometric functions are widely used in solving real-world problems as well as in the development of mathematics. Whatever their
use, it is often of value to be able to change a trigonometric
expression from one form to an equivalent more useful form. This
involves the use of identities. Recall that an equation in one or more
variables is said to be an identity if the left side is equal to the right
side for all replacements of the variables for which both sides are
defined. For example, the equation
T
x2 ⫺ 2x ⫺ 8 ⫽ (x ⫺ 4)(x ⫹ 2)
is an identity, but
x2 ⫺ 2x ⫺ 8 ⫽ 0
is not. The latter is called a conditional equation, since it holds only
for certain values of x and not for all values for which both sides are
defined. The first four sections of the chapter deal with trigonometric identities and the last section with conditional trigonometric
equations.
Preparing for This Chapter
Before getting started on this chapter, review the following concepts:
Operations on Polynomials (Appendix A, Sections 2 and 3)
Rational Expressions (Appendix A, Section 4)
Cartesian Coordinate System (Chapter 1, Section 1)
Quadratic Equations (Chapter 2, Section 5)
Basic Identities (Chapter 5, Section 2)
Section 6-1 Basic Identities and Their Use
Basic Identities
Establishing Other Identities
In this section we review the basic identities introduced in Section 5-2 and show
how they are used to verify other identities.
Basic Identities
In the following box we list for convenient reference the basic identities introduced in Section 5-2. These identities will be used very frequently in the work
that follows and should be memorized.
452
6-1 Basic Identities and Their Use
453
BASIC TRIGONOMETRIC IDENTITIES
Reciprocal Identities
csc x ⫽
1
sin x
sec x ⫽
1
cos x
cot x ⫽
cos x
sin x
cot x ⫽
1
tan x
Quotient Identities
tan x ⫽
sin x
cos x
Identities for Negatives
sin (⫺x) ⫽ ⫺sin x
cos (⫺x) ⫽ cos x
tan (⫺x) ⫽ ⫺tan x
Pythagorean Identities
sin2 x ⫹ cos2 x ⫽ 1
tan2 x ⫹ 1 ⫽ sec2 x
1 ⫹ cot2 x ⫽ csc2 x
All these identities were established in Section 5-2 (the second and third
Pythagorean identities were established in Problems 87 and 88 in Exercise 5-2).
Explore/Discuss
1
Discuss an easy way to recall the second and third Pythagorean identities
from the first. [Hint: Divide through the first Pythagorean identity by
appropriate expressions.]
Establishing Other Identities
As indicated above, when working with trigonometric expressions it is often desirable to convert one form to an equivalent form that may be more useful. This
section is designed to give you experience in this process. In addition to using
the basic identities and other verified identities, we will often use basic algebraic
operations such as multiplication, factoring, combining and reducing fractions,
and so on. The following examples illustrate some of the techniques used to verify certain identities. The steps illustrated are not necessarily unique—often, there
is more than one path to a desired goal. To become proficient in the use of identities, it is important that you work out many problems on your own.
EXAMPLE
1
Identity Verification
Verify the identity cos x tan x ⫽ sin x.
454
6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS
Verification
Generally, we proceed by starting with the more complicated of the two sides,
and transform that side into the other side in one or more steps using basic identities, algebra, or other established identities. Thus,
cos x tan x ⫽ cos x
sin x
cos x
⫽ sin x
MATCHED PROBLEM
Quotient identity
Algebra
Verify the identity sin x cot x ⫽ cos x.
1
Explore/Discuss
2
EXAMPLE
2
Verification
Graph the left and right sides of the identity in Example 1 in a graphing
utility by letting y1 ⫽ cos x tan x and y2 ⫽ sin x. Use TRACE, moving
back and forth between the graphs of y1 and y2, to compare values of y
for given values of x. What does this investigation illustrate?
Identity Verification
Verify the identity sec (⫺x) ⫽ sec x.
sec (⫺x) ⫽
⫽
1
cos (⫺x)
Reciprocal identity
1
cos x
Identity for negatives
⫽ sec x
MATCHED PROBLEM
Reciprocal identity
Verify the identity csc (⫺x) ⫽ ⫺csc x.
2
EXAMPLE
3
Verification
Identity Verification
Verify the identity cot x cos x ⫹ sin x ⫽ csc x.
cos x
cos x ⫹ sin x
sin x
Quotient identity
⫽
cos2 x
⫹ sin x
sin x
Algebra
⫽
cos2 x ⫹ sin2 x
sin x
Algebra
⫽
1
sin x
Pythagorean identity
cot x cos x ⫹ sin x ⫽
⫽ csc x
Reciprocal identity
6-1 Basic Identities and Their Use
455
Key Algebraic Steps in Example 3
a
a2
a2 ⫹ b2
a⫹b⫽
⫹b⫽
b
b
b
MATCHED PROBLEM
Verify the identity tan x sin x ⫹ cos x ⫽ sec x.
3
To verify an identity, proceed from one side to the other, or from both sides
to the middle, making sure all steps are reversible. Do not use properties of equality to perform the same operation on both sides of the equation. Even though
there is no fixed method of verification that works for all identities, there are certain steps that help in many cases.
SUGGESTED STEPS IN VERIFYING IDENTITIES
1. Start with the more complicated side of the identity, and transform it
into the simpler side.
2. Try algebraic operations such as multiplying, factoring, combining
fractions, and splitting fractions.
3. If other steps fail, express each function in terms of sine and cosine
functions, and then perform appropriate algebraic operations.
4. At each step, keep the other side of the identity in mind. This often
reveals what you should do in order to get there.
EXAMPLE
4
Verification
Identity Verification
Verify the identity
cos x
1 ⫹ sin x
⫹
⫽ 2 sec x.
cos x
1 ⫹ sin x
1 ⫹ sin x
cos x
(1 ⫹ sin x)2 ⫹ cos2 x
⫹
⫽
cos x
1 ⫹ sin x
cos x (1 ⫹ sin x)
Algebra
⫽
1 ⫹ 2 sin x ⫹ sin2 x ⫹ cos2 x
cos x (1 ⫹ sin x)
⫽
1 ⫹ 2 sin x ⫹ 1
cos x (1 ⫹ sin x)
⫽
2 ⫹ 2 sin x
cos x (1 ⫹ sin x)
Algebra
⫽
2(1 ⫹ sin x)
cos x (1 ⫹ sin x)
Algebra
⫽
2
cos x
Algebra
⫽ 2 sec x
Algebra
Pythagorean
identity
Reciprocal
identity
456
6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS
Key Algebraic Steps in Example 4
a
b a2 ⫹ b2
⫹ ⫽
(1 ⫹ c)2 ⫽ 1 ⫹ 2c ⫹ c2
b
a
ba
MATCHED PROBLEM
4
EXAMPLE
5
Verification
Verify the identity
m(a ⫹ b) m
⫽
n(a ⫹ b)
n
sin x
1 ⫹ cos x
⫹
⫽ 2 csc x.
sin x
1 ⫹ cos x
Identity Verification
sin2 x ⫹ 2 sin x ⫹ 1 1 ⫹ sin x
⫽
.
cos2 x
1 ⫺ sin x
sin2 x ⫹ 2 sin x ⫹ 1 (sin x ⫹ 1)2
⫽
Algebra
cos2 x
cos2 x
Verify the identity
⫽
(sin x ⫹ 1)2
1 ⫺ sin2 x
Pythagorean identity
⫽
(1 ⫹ sin x)2
(1 ⫺ sin x)(1 ⫹ sin x)
Algebra
⫽
1 ⫹ sin x
1 ⫺ sin x
Algebra
Key Algebraic Steps in Example 5
a2 ⫹ 2a ⫹ 1 ⫽ (a ⫹ 1)2
1 ⫺ b2 ⫽ (1 ⫺ b)(1 ⫹ b)
MATCHED PROBLEM
Verify the identity sec4 x ⫺ 2 sec2 x tan2 x ⫹ tan4 x ⫽ 1.
5
EXAMPLE
6
Verification
Identity Verification
tan x ⫺ cot x
⫽ 1 ⫺ 2 cos2 x.
Verify the identity
tan x ⫹ cot x
sin x
cos x
⫺
tan x ⫺ cot x cos x
sin x
⫽
tan x ⫹ cot x
sin x
cos x
⫹
cos x
sin x
Change to sines and
cosines (quotient
identities).
⫺
冢
cos x
sin x 冣
⫽
sin x
cos x
(sin x)(cos x)冢
⫹
cos x
sin x 冣
(sin x)(cos x)
sin x
⫽
sin2 x ⫺ cos2 x
sin2 x ⫹ cos2 x
⫽
1 ⫺ cos2 x ⫺ cos2 x
1
⫽ 1 ⫺ 2 cos2 x
cos x
Multiply numerator
and denominator by
(sin x)(cos x), and use
algebra to transform
the compound fraction
into a simple fraction.
Pythagorean identity
Algebra
6-1 Basic Identities and Their Use
457
Key Algebraic Steps in Example 6
a b
a b
⫺
ab ⫺
b a
b a
a2 ⫺ b2
⫽
⫽ 2
a b
a b
a ⫹ b2
⫹
ab ⫹
b a
b a
冢
冢
冣
冣
Verify the identity cot x ⫺ tan x ⫽
MATCHED PROBLEM
6
2 cos2 x ⫺ 1
.
sin x cos x
Just observing how others verify identities won’t make you good at
it. You must verify a large number on your own. With practice the
process will seem less complicated.
EXAMPLE
Testing Identities Using a Graphing Utility
7
Use a graphing utility to test whether each of the following is an identity. If
an equation appears to be an identity, verify it. If the equation does not appear
to be an identity, find a value of x for which both sides are defined but are
not equal.
(A) tan x ⫹ 1 ⫽ (sec x)(sin x ⫺ cos x)
(B) tan x ⫺ 1 ⫽ (sec x)(sin x ⫺ cos x)
(A) Graph each side of the equation in the same viewing window (Fig. 1). The
equation is not an identity, since the graphs do not match.
Solutions
FIGURE 1
4
⫺2␲
2␲
⫺4
Try x ⫽ 0.
Left side: tan 0 ⫹ 1 ⫽ 1
Right side: (sec 0)(sin 0 ⫺ cos 0) ⫽ ⫺1
Finding one value of x for which both sides are defined, but are not equal,
is enough to verify that the equation is not an identity.
(B) Graph each side of the equation in the same viewing window (Fig. 2). The
equation appears to be an identity, which we now verify:
FIGURE 2
4
(sec x) (sin x ⫺ cos x)
⫺2␲
2␲
⫺4
冢cos1 x 冣(sin x ⫺ cos x)
sin x
cos x
⫺
⫽冢
cos x cos x 冣
⫽
⫽ tan x ⫺ 1
458
6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS
MATCHED PROBLEM
7
Use a graphing utility to test whether each of the following is an identity. If an
equation appears to be an identity, verify it. If the equation does not appear to be
an identity, find a value of x for which both sides are defined but are not equal.
sin x
sin x
(A)
⫽ csc x
(B)
⫽ sec x
2
1 ⫺ cos x
1 ⫺ cos2 x
Answers to Matched Problems
In the following identity verifications, other correct sequences of steps are possible—the process is not unique.
cos x
1
1
1. sin x cot x ⫽ sin x
2. csc (⫺x) ⫽
⫽ cos x
⫽
⫽ ⫺csc x
sin x
sin (⫺x) ⫺sin x
sin2 x ⫹ cos2 x
1
sin2 x
3. tan x sin x ⫹ cos x ⫽
⫹ cos x ⫽
⫽
⫽ sec x
cos x
cos x
cos x
1 ⫹ cos x
sin x
(1 ⫹ cos x)2 ⫹ sin2 x 1 ⫹ 2 cos x ⫹ cos2 x ⫹ sin2 x
2(1 ⫹ cos x)
4.
⫹
⫽
⫽
⫽
⫽ 2 csc x
sin x
1 ⫹ cos x
sin x (1 ⫹ cos x)
sin x (1 ⫹ cos x)
sin x (1 ⫹ cos x)
5. sec4 x ⫺ 2 sec2 x tan2 x ⫹ tan4 x ⫽ (sec2 x ⫺ tan2 x)2 ⫽ 12 ⫽ 1
cos x
sin x
cos2 x ⫺ sin2 x cos2 x ⫺ (1 ⫺ cos2 x) 2 cos2 x ⫺ 1
6. cot x ⫺ tan x ⫽
⫺
⫽
⫽
⫽
sin x
cos x
sin x cos x
sin x cos x
sin x cos x
7. (A) An identity:
4
⫺2␲
2␲
⫺4
sin x
1
sin x
⫽
⫽
⫽ csc x
1 ⫺ cos2 x sin2 x sin x
(B) Not an identity: the left side is not equal to the right side for x ⫽ 1, for example.
4
⫺2␲
2␲
⫺4
6-1 Basic Identities and Their Use
459
29. ⫺␲ ⱕ x ⱕ ␲
EXERCISE 6-1
(A) y ⫽
A
cos x
cot x sin x
(B) y ⫽ 1
30. ⫺␲ ⱕ x ⱕ ␲
Verify that Problems 1–26 are identities.
1. sin ␪ sec ␪ ⫽ tan ␪
2. cos ␪ csc ␪ ⫽ cot ␪
3. cot u sec u sin u ⫽ 1
4. tan ␪ csc ␪ cos ␪ ⫽ 1
5.
sin (⫺x)
⫽ ⫺tan x
cos (⫺x)
7. sin ␣ ⫽
6. cot (⫺x) tan x ⫽ ⫺1
tan ␣ cot ␣
csc ␣
8. tan ␣ ⫽
cos ␣ sec ␣
cot ␣
9. cot u ⫹ 1 ⫽ (csc u)(cos u ⫹ sin u)
10. tan u ⫹ 1 ⫽ (sec u)(sin u ⫹ cos u)
11.
cos x ⫺ sin x
⫽ csc x ⫺ sec x
sin x cos x
12.
cos x ⫺ sin x
⫽ cot x ⫺ tan x
sin x cos x
2
13.
15.
(A) y ⫽
sin x
cos x tan x
B
Verify that Problems 31–60 are identities.
31.
1 ⫺ (sin x ⫺ cos x)2
⫽ 2 cos x
sin x
32.
1 ⫺ cos2 y
⫽ tan2 y
(1 ⫺ sin y)(1 ⫹ sin y)
33. cos ␪ ⫹ sin ␪ ⫽
cot ␪ ⫹ 1
csc ␪
34. sin ␪ ⫹ cos ␪ ⫽
tan ␪ ⫹ 1
sec ␪
2
sin2 t
⫹ cos t ⫽ sec t
cos t
14.
cos x
⫽ sec x
1 ⫺ sin2 x
16.
cos2 t
⫹ sin t ⫽ csc t
sin t
sin u
⫽ csc u
1 ⫺ cos2 u
17. (1 ⫺ cos u)(1 ⫹ cos u) ⫽ sin2 u
20. (sin x ⫹ cos x)2 ⫽ 1 ⫹ 2 sin x cos x
21. (sec t ⫹ 1)(sec t ⫺ 1) ⫽ tan2 t
cos2 y
1 ⫹ sin y
39.
csc ␪
⫽ cos ␪
cot ␪ ⫹ tan ␪
43. ln (cot x) ⫽ ⫺ln (tan x)
24. sec2 u ⫺ tan2 u ⫽ 1
45.
cos x ⫹ tan x
25. cot x ⫹ sec x ⫽
sin x
26. sin m (csc m ⫺ sin m) ⫽ cos2 m
In Problems 27–30, graph all parts of each problem in the
same viewing window in a graphing utility.
27. ⫺␲ ⱕ x ⱕ ␲
(B) y ⫽ cos x
2
(C) y ⫽ sin2 x ⫹ cos2 x
28. ⫺␲ ⱕ x ⱕ ␲
(B) y ⫽ tan2 x
(C) y ⫽ sec x ⫺ tan x
2
40.
1 ⫹ sec ␪
⫽ csc ␪
sin ␪ ⫹ tan ␪
42. ln (cot x) ⫽ ln (cos x) ⫺ ln (sin x)
23. csc2 x ⫺ cot2 x ⫽ 1
2
36. 1 ⫺ sin y ⫽
41. ln (tan x) ⫽ ln (sin x) ⫺ ln (cos x)
22. (csc t ⫺ 1)(csc t ⫹ 1) ⫽ cot2 t
(A) y ⫽ sec2 x
1 ⫹ cos y
sin2 y
⫽
1 ⫺ cos y (1 ⫺ cos y)2
38. sec2 x ⫹ csc2 x ⫽ sec2 x csc2 x
19. cos2 x ⫺ sin2 x ⫽ 1 ⫺ 2 sin2 x
(A) y ⫽ sin x
35.
37. tan2 x ⫺ sin2 x ⫽ tan2 x sin2 x
18. (1 ⫺ sin t)(1 ⫹ sin t) ⫽ cos2 t
2
(B) y ⫽ 1
1 ⫺ cos A sec A ⫺ 1
⫽
1 ⫹ cos A sec A ⫹ 1
44. ln (csc x) ⫽ ⫺ln (sin x)
46.
1 ⫺ csc y sin y ⫺ 1
⫽
1 ⫹ csc y sin y ⫹ 1
47. sin4 w ⫺ cos4 w ⫽ 1 ⫺ 2 cos2 w
48. sin4 x ⫹ 2 sin2 x cos2 x ⫹ cos4 x ⫽ 1
49. sec x ⫺
cos x
⫽ tan x
1 ⫹ sin x
50. csc n ⫺
sin n
⫽ cot n
1 ⫹ cos n
51.
cos2 z ⫺ 3 cos z ⫹ 2 2 ⫺ cos z
⫽
sin2 z
1 ⫹ cos z
52.
sin2 t ⫹ 4 sin t ⫹ 3 3 ⫹ sin t
⫽
cos2 t
1 ⫺ sin t
460
6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS
53.
cos3 ␪ ⫺ sin3 ␪
⫽ 1 ⫹ sin ␪ cos ␪
cos ␪ ⫺ sin ␪
77. tan ␣ ⫹ cot ␤ ⫽
54.
cos3 u ⫹ sin3 u
⫽ 1 ⫺ sin u cos u
cos u ⫹ sin u
78.
55. (sec x ⫺ tan x)2 ⫽
1 ⫺ sin x
1 ⫹ sin x
56. (cot u ⫺ csc u)2 ⫽
1 ⫺ cos u
1 ⫹ cos u
tan ␤ ⫹ cot ␣
tan ␤ cot ␣
tan ␣ ⫹ tan ␤
cot ␣ ⫹ cot ␤
⫽
cot ␣ cot ␤ ⫺ 1 1 ⫺ tan ␣ tan ␤
From the graph of y1 ⫽ f(x) in a graphing utility, find a simpler
function of the form g(x) ⫽ k ⫹ AT(x), where T(x) is one of the
six trigonometric functions that has the same graph as y1 ⫽
f(x). Verify the identity f(x) ⫽ g(x).
57.
csc4 x ⫺ 1
⫽ 2 ⫹ cot2 x
cot2 x
58.
sec4 x ⫺ 1
⫽ 2 ⫹ tan2 x
tan2 x
79. f(x) ⫽
1 ⫺ sin2 x
⫹ sin x cos x
tan x
59.
1 ⫹ sin v
cos v
⫽
cos v
1 ⫺ sin v
60.
1 ⫹ cos x
sin x
⫽
1 ⫺ cos x
sin x
80. f(x) ⫽
1 ⫹ sin x
cos x
⫺
2 cos x
2 ⫹ 2 sin x
81. f(x) ⫽
cos2 x
1 ⫹ sin x ⫺ cos2 x
82. f(x) ⫽
tan x sin x
1 ⫺ cos x
83. f(x) ⫽
1 ⫹ cos x ⫺ 2 cos2 x
sin2 x
⫺
1 ⫺ cos x
1 ⫹ cos x
84. f(x) ⫽
3 sin x ⫺ 2 sin x cos x 1 ⫹ cos x
⫺
1 ⫺ cos x
sin x
Use a graphing utility to test whether each of the following is
an identity. If an equation appears to be an identity, verify it. If
the equation does not appear to be an identity, find a value of x
for which both sides are defined but are not equal.
61.
cos (⫺x)
sin (⫺x)
⫽ ⫺1 62.
⫽1
cos (⫺x) tan (⫺x)
sin x cot (⫺x)
63.
sin x
⫽ ⫺1
cos x tan (⫺x)
64.
cos x
⫽1
sin (⫺x) cot (⫺x)
65. sin x ⫹
cos2 x
⫽ sec x
sin x
66.
1 ⫺ tan2 x
⫽ tan2 x
1 ⫺ cot2 x
67. sin x ⫹
cos2 x
⫽ csc x
sin x
68.
tan2 x ⫺ 1
⫽ tan2 x
1 ⫺ cot2 x
tan x
1
⫽
69.
sin x ⫺ 2 tan x cos x ⫺ 2
70.
cos x
cos x
⫹
⫽ 2 sec x
1 ⫺ sin x 1 ⫹ sin x
71.
tan x
1
⫽
sin x ⫹ 2 tan x cos x ⫺ 2
72.
cos x
cos x
⫺
⫽ 2 csc x
sin x ⫹ 1 sin x ⫺ 1
C
Verify that Problems 73–78 are identities.
73.
2 sin2 x ⫹ 3 cos x ⫺ 3 2 cos x ⫺ 1
⫽
sin2 x
1 ⫹ cos x
74.
3 cos2 z ⫹ 5 sin z ⫺ 5 3 sin z ⫺ 2
⫽
cos2 z
1 ⫹ sin z
75.
tan u ⫹ sin u sec u ⫹ 1
⫺
⫽0
tan u ⫺ sin u sec u ⫺ 1
tan x ⫹ tan y
sin x cos y ⫹ cos x sin y
⫽
76.
cos x cos y ⫺ sin x sin y 1 ⫺ tan x tan y
Each of the equations in Problems 85–92 is an identity in
certain quadrants associated with x. Indicate which quadrants.
85. 兹1 ⫺ cos2 x ⫽ ⫺sin x
86. 兹1 ⫺ sin2 x ⫽ cos x
87. 兹1 ⫺ cos2 x ⫽ sin x
88. 兹1 ⫺ sin2 x ⫽ ⫺cos x
89. 兹1 ⫺ sin2 x ⫽ 兩cos x兩
90. 兹1 ⫺ cos2 x ⫽ 兩sin x兩
91.
sin x
⫽ tan x
兹1 ⫺ sin2 x
92.
sin x
⫽ ⫺tan x
兹1 ⫺ sin2 x
In calculus, trigonometric substitutions provide an effective
way to rationalize the radical forms 兹a2 ⫺ u2 and 兹a2 ⫹ u2,
which in turn leads to the solution to an important class of
problems. Problems 93–96 involve such transformations.
[Recall: 兹x2 ⫽ 兩x兩 for all real numbers x.]
93. In the radical form 兹a2 ⫺ u2, a ⬎ 0, let u ⫽ a sin x,
⫺␲/2 ⬍ x ⬍ ␲/2. Simplify, using a basic identity, and
write the final form free of radicals.
94. In the radical form 兹a2 ⫺ u2, a ⬎ 0, let u ⫽ a cos x,
0 ⬍ x ⬍ ␲. Simplify, using a basic identity, and write the
final form free of radicals.
95. In the radical form 兹a2 ⫹ u2, a ⬎ 0, let u ⫽ a tan x,
0 ⬍ x ⬍ ␲/2. Simplify, using a basic identity, and write
the final form free of radicals.
96. In the radical form 兹a2 ⫹ u2, a ⬎ 0, let u ⫽ a cot x,
0 ⬍ x ⬍ ␲/2. Simplify, using a basic identity, and write
the final form free of radicals.