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Analytic Trigonometry
Using Fundamental Identities
What you should learn • Recognize and write the fundamental trigonometric identities. • Use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions.
Introduction In Chapter 4, you studied the basic definitions, properties, graphs, and applications of the individual trigonometric functions. In this chapter, you will learn how to use the fundamental identities to do the following. 1. 2. 3. 4.
Evaluate trigonometric functions. Simplify trigonometric expressions. Develop additional trigonometric identities. Solve trigonometric equations.
Why you should learn it Fundamental trigonometric identities can be used to simplify trigonometric expressions. For instance, in Exercise 99 on page 381, you can use trigonometric identities to simplify an expression for the coefficient of friction.
Fundamental Trigonometric Identities Reciprocal Identities sin u
1 csc u
cos u
1 sec u
tan u
1 cot u
csc u
1 sin u
sec u
1 cos u
cot u
1 tan u
cot u
cos u sin u
Quotient Identities tan u
sin u cos u
Pythagorean Identities sin2 u cos 2 u 1
1 tan2 u sec 2 u
1 cot 2 u csc 2 u
Cofunction Identities sin
2 u cos u
tan
2 u cot u
sec
2 u csc u
cos cot
2 u sin u
2 u tan u
csc
2 u sec u
Even/Odd Identities sinu sin u
cosu cos u
tanu tan u
cscu csc u
secu sec u
cotu cot u
Pythagorean identities are sometimes used in radical form such as sin u ± 1 cos 2 u The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter.
or tan u ± sec 2 u 1 where the sign depends on the choice of u.
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Using Fundamental Identities
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Using the Fundamental Identities You should learn the fundamental trigonometric identities well, because they are used frequently in trigonometry and they will also appear later in calculus. Note that u can be an angle, a real number, or a variable.
One common use of trigonometric identities is to use given values of trigonometric functions to evaluate other trigonometric functions.
Example 1
Using Identities to Evaluate a Function
3 Use the values sec u 2 and tan u > 0 to find the values of all six trigonometric functions.
Solution Using a reciprocal identity, you have 1 1 2 . sec u 32 3
cos u
Using a Pythagorean identity, you have sin2 u 1 cos 2 u
3
1
Te c h n o l o g y You can use a graphing utility to check the result of Example 2. To do this, graph y1 sin x
cos 2
x sin x
and y2 sin3 x
1
2
Substitute 3 for cos u.
4 5 . 9 9
Simplify.
Because sec u < 0 and tan u > 0, it follows that u lies in Quadrant III. Moreover, because sin u is negative when u is in Quadrant III, you can choose the negative root and obtain sin u 53. Now, knowing the values of the sine and cosine, you can find the values of all six trigonometric functions. sin u
in the same viewing window, as shown below. Because Example 2 shows the equivalence algebraically and the two graphs appear to coincide, you can conclude that the expressions are equivalent.
2
Pythagorean identity 2
5
cos u tan u
3 2 3
sin u 53 5 cos u 23 2
csc u
1 3 35 sin u 5 5
sec u
1 3 cos u 2
cot u
1 2 25 tan u 5 5
Now try Exercise 11.
2
Example 2 −π
Simplifying a Trigonometric Expression
π
Simplify sin x cos 2 x sin x. −2
Solution First factor out a common monomial factor and then use a fundamental identity. sin x cos 2 x sin x sin xcos2 x 1
Remind students that they must use an algebraic approach to prove that two expressions are equivalent. A graphical approach can only confirm that the simplification found using algebraic techniques is correct.
sin x1
cos 2
sin xsin2 x
sin3
x
Now try Exercise 45.
Factor out common monomial factor.
x
Factor out 1. Pythagorean identity Multiply.
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Analytic Trigonometry
When factoring trigonometric expressions, it is helpful to find a special polynomial factoring form that fits the expression, as shown in Example 3.
Example 3
Factoring Trigonometric Expressions
Factor each expression. a. sec 2 1
b. 4 tan2 tan 3
Solution a. Here you have the difference of two squares, which factors as sec2 1 sec 1sec 1). b. This expression has the polynomial form ax 2 bx c, and it factors as 4 tan2 tan 3 4 tan 3tan 1. Now try Exercise 47. On occasion, factoring or simplifying can best be done by first rewriting the expression in terms of just one trigonometric function or in terms of sine and cosine only. These strategies are illustrated in Examples 4 and 5, respectively.
Example 4
Factoring a Trigonometric Expression
Factor csc 2 x cot x 3.
Solution Use the identity csc 2 x 1 cot 2 x to rewrite the expression in terms of the cotangent. csc 2 x cot x 3 1 cot 2 x cot x 3
cot 2
x cot x 2
cot x 2cot x 1
Pythagorean identity Combine like terms. Factor.
Now try Exercise 51.
Example 5
Simplifying a Trigonometric Expression
Simplify sin t cot t cos t.
Solution Remember that when adding rational expressions, you must first find the least common denominator (LCD). In Example 5, the LCD is sin t.
Begin by rewriting cot t in terms of sine and cosine. sin t cot t cos t sin t
sin t cos t cos t
sin2 t cos 2 t sin t 1 sin t
csc t Now try Exercise 57.
Quotient identity
Add fractions. Pythagorean identity Reciprocal identity
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Section 5.1
Using Fundamental Identities
377
Adding Trigonometric Expressions
Example 6
Perform the addition and simplify. sin cos 1 cos sin
Solution sin cos sin sin (cos 1 cos 1 cos sin 1 cos sin sin2 cos2 cos 1 cos sin 1 cos 1 cos sin
1 sin
Multiply. Pythagorean identity: sin2 cos2 1 Divide out common factor.
csc
Reciprocal identity
Now try Exercise 61. The last two examples in this section involve techniques for rewriting expressions in forms that are used in calculus.
Example 7 Rewrite
Rewriting a Trigonometric Expression
1 so that it is not in fractional form. 1 sin x
Solution From the Pythagorean identity cos 2 x 1 sin2 x 1 sin x1 sin x, you can see that multiplying both the numerator and the denominator by 1 sin x will produce a monomial denominator. 1 1 1 sin x 1 sin x
1 sin x
1 sin x
Multiply numerator and denominator by 1 sin x.
1 sin x 1 sin2 x
Multiply.
1 sin x cos 2 x
Pythagorean identity
1 sin x cos 2 x cos 2 x
Write as separate fractions.
1 sin x cos 2 x cos x
1
cos x
sec2 x tan x sec x Now try Exercise 65.
Product of fractions Reciprocal and quotient identities
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Analytic Trigonometry
Trigonometric Substitution
Example 8
Use the substitution x 2 tan , 0 < < 2, to write 4 x 2 Activities 1. Simplify, using the fundamental trigonometric identities. cot2 csc2 Answer: cos2 2. Use the trigonometric substitution x 4 sec to rewrite the expression x2 16 as a trigonometric function of , where 0 < < . 2 Answer: 4 tan
4+
2
x
θ = arctan x 2 2 x Angle whose tangent is . 2 FIGURE 5.1
x
as a trigonometric function of .
Solution Begin by letting x 2 tan . Then, you can obtain 4 x 2 4 2 tan 2
Substitute 2 tan for x.
4 4 tan2
Rule of exponents
41 tan2
Factor.
4 sec 2
Pythagorean identity
2 sec .
sec > 0 for 0 < < 2
Now try Exercise 77. Figure 5.1 shows the right triangle illustration of the trigonometric substitution x 2 tan in Example 8. You can use this triangle to check the solution of Example 8. For 0 < < 2, you have opp x,
adj 2,
and hyp 4 x 2 .
With these expressions, you can write the following. sec sec
hyp adj 4 x 2
2
2 sec 4 x 2 So, the solution checks.
Example 9
Rewriting a Logarithmic Expression
Rewrite ln csc ln tan as a single logarithm and simplify the result.
Solution
ln csc ln tan ln csc tan
sin
ln
1 sin
ln
1 cos
ln sec
cos
Now try Exercise 91.
Product Property of Logarithms Reciprocal and quotient identities
Simplify. Reciprocal identity
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Section 5.1
5.1
Exercises
Using Fundamental Identities
379
The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help.
VOCABULARY CHECK: Fill in the blank to complete the trigonometric identity. 1.
sin u ________ cos u
2.
1 ________ sec u
3.
1 ________ tan u
4.
1 ________ sin u
6. 1 tan2 u ________
5. 1 ________ csc2 u 7. sin
2 u ________
8. sec
9. cosu ________
2 u ________
10. tanu ________
PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–14, use the given values to evaluate (if possible) all six trigonometric functions. 1. sin x 2. tan x
3
2
, cos x
3
3
,
cos x sin
3. sec 2, 4. 5.
csc 53, 5 tan x 12 ,
tan
1 2 2 2
3
1 9. sinx , 3
2
23.
sec4
x
tan4
(c) sin2 x (f) sec2 x tan2 x
x
22. cos2 xsec2 x 1 x
sec2 x 1 25. sin2 x
10
24. cot x sec x 26.
cos22 x cos x
10
cos x tan x
In Exercises 27–44, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.
4 5 2
27. cot sec
28. cos tan
29. sin csc sin
30. sec 2 x1 sin2 x
4
11. tan 2, sin < 0 12. csc 5, cos < 0 cot 0
14. tan is undefined,
(e)
sec2
21. sin x sec x
10. sec x 4, sin x > 0
13. sin 1,
sin2 x cos2 x
(b) tan x
(d) sin x tan x
sec x 13 12
2 x 5,
20.
(a) csc x
3 35 7. sec , csc 2 5 8. cos
sinx cosx
In Exercises 21–26, match the trigonometric expression with one of the following.
3
3 4
6. cot 3, sin
19.
31.
cot x csc x
32.
csc sec
33.
1 sin2 x csc2 x 1
34.
1 tan2 x 1
36.
tan2 sec2
35. sec
sin > 0
In Exercises 15–20, match the trigonometric expression with one of the following. (a) sec x
(b) 1
(c) cot x
(d) 1
(e) tan x
(f) sin x
37. cos 39.
sin
tan
2 xsec x
cos2 y 1 sin y
41. sin tan cos
15. sec x cos x
16. tan x csc x
43. cot u sin u tan u cos u
17. cot2 x csc 2 x
18. 1 cos 2 xcsc x
44. sin sec cos csc
38. cot
2 xcos x
40. cos t1 tan2 t 42. csc tan sec
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Analytic Trigonometry
In Exercises 45–56, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer. 45.
tan2
x
tan2
x
sin2
46.
x
47. sin2 x sec2 x sin2 x 49.
sin2
x
csc2
x
sin2
50.
cos2 x 4 cos x 2
51. tan4 x 2 tan2 x 1
52. 1 2 cos2 x cos4 x
53. sin4 x cos4 x
54. sec4 x tan4 x
55. csc3 x csc2 x csc x 1
In Exercises 57– 60, perform the multiplication and use the fundamental identities to simplify. There is more than one correct form of each answer. 57. sin x cos x2 58. cot x csc xcot x csc x
73. cos x cot x sin x 74. sec x csc x tan x
60. 3 3 sin x3 3 sin x In Exercises 61–64, perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer. 1 1 sec x 1 sec x 1
62.
63.
cos x 1 sin x 1 sin x cos x
64. tan x
sec2 x tan x
In Exercises 65– 68, rewrite the expression so that it is not in fractional form. There is more than one correct form of each answer.
67.
5 66. tan x sec x
3 sec x tan x
68.
0.2
0.4
0.6
0.8
x csc x 1
1.0
y1
76.
cos 1 1 sin 2 cos 1 sin
In Exercises 77– 82, use the trigonometric substitution to write the algebraic expression as a trigonometric function of , where 0 < < /2. x 3 cos x 2 cos
79. x 2 9,
x 3 sec
80. x 2 4,
x 2 sec
81. x 2 25,
x 5 tan
82. x 2 100,
x 10 tan
In Exercises 83– 86, use the trigonometric substitution to write the algebraic equation as a trigonometric function of , where /2 < < /2. Then find sin and cos . 83. 3 9 x 2, 84. 3 36 x 2,
x 3 sin x 6 sin
85. 22 16 4x 2,
x 2 cos
86. 53 100 x 2,
x 10 cos
87. sin 1 cos2
1.4
88. cos 1 sin2 89. sec 1 tan2 90. csc 1 cot2 In Exercises 91–94, rewrite the expression as a single logarithm and simplify the result.
lncot t ln1 tan2 t
91. ln cos x ln sin x
y2 69. y1 cos
1.2
1 1 cos x sin x cos x
In Exercises 87–90, use a graphing utility to solve the equation for , where 0 ≤ < 2.
tan2
Numerical and Graphical Analysis In Exercises 69 –72, use a graphing utility to complete the table and graph the functions. Make a conjecture about y1 and y2. x
75.
78. 64 16x 2,
1 1 1 cos x 1 cos x
y2 tan2 x tan4 x
In Exercises 73–76, use a graphing utility to determine which of the six trigonometric functions is equal to the expression. Verify your answer algebraically.
77. 9 x 2,
59. 2 csc x 22 csc x 2
61.
1 sin x cos x
y2
72. y1 sec4 x sec2 x,
56. sec3 x sec2 x sec x 1
sin2 y 65. 1 cos y
cos x , 1 sin x
x
48. cos2 x cos2 x tan2 x
sec2 x 1 sec x 1
71. y1
92. ln sec x ln sin x
x , 2
70. y1 sec x cos x,
93. y2 sin x y2 sin x tan x
94. lncos2 t ln1 tan2 t
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Section 5.1 In Exercises 95–98, use a calculator to demonstrate the identity for each value of . 95. csc2 cot2 1 (a) 132 ,
(b)
2 7
97. cos
105. As x →
(b) 3.1
(b) 0.8
In Exercises 107–112, determine whether or not the equation is an identity, and give a reason for your answer.
98. sin sin (a) 250 ,
, tan x → and cot x → . 2
106. As x → , sin x → and csc x → .
2 sin
(a) 80 ,
In Exercises 103–106, fill in the blanks. (Note: The notation x → c indicates that x approaches c from the right and x → c indicates that x approaches c from the left.) 103. As x → , sin x → and csc x → . 2 104. As x → 0 , cos x → and sec x → .
96. tan2 1 sec2 (a) 346 ,
381
Using Fundamental Identities
1 (b) 2
107. cos 1 sin2
99. Friction The forces acting on an object weighing W units on an inclined plane positioned at an angle of with the horizontal (see figure) are modeled by
W cos W sin where is the coefficient of friction. Solve the equation for and simplify the result.
108. cot csc2 1
sin k tan , k is a constant. cos k 1 110. 5 sec 5 cos 111. sin csc 1 112. csc2 1 109.
113. Use the definitions of sine and cosine to derive the Pythagorean identity sin2 cos2 1. 114. Writing Use the Pythagorean identity sin2 cos2 1
W
to derive the other Pythagorean identities, 1 tan2 sec2 and 1 cot2 csc2 . Discuss how to remember these identities and other fundamental identities.
θ
Skills Review 100. Rate of Change
The rate of change of the function
f x csc x sin x is given by the expression csc x cot x cos x. Show that this expression can also be written as cos x cot2 x.
In Exercises 115 and 116, perform the operation and simplify. 115. x 5x 5
117. 119.
101. The even and odd trigonometric identities are helpful for determining whether the value of a trigonometric function is positive or negative. 102. A cofunction identity can be used to transform a tangent function so that it can be represented by a cosecant function.
2
In Exercises 117–120, perform the addition or subtraction and simplify.
Synthesis True or False? In Exercises 101 and 102, determine whether the statement is true or false. Justify your answer.
116. 2z 3
1 x x5 x8 x2
118.
2x 7 4 x4
120.
6x 3 x4 4x x2
x x2 25 x 5
In Exercises 121–124, sketch the graph of the function. (Include two full periods.) 121. f x
1 sin x 2
123. f x
1 sec x 2 4
122. f x 2 tan
124. f x
x 2
3 cosx 3 2
