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 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

Fundamental trigonometric identities can be used to simplify trigonometric expressions. For instance, in Exercise 99, you can use trigonometric identities to simplify an expression for the coefficient of friction.

Reciprocal Identities 1 1 sin u 5 cos u 5 csc u sec u csc u 5

1 sin u

sec u 5

1 cos u

cot u 5

cos u sin u

Quotient Identities sin u tan u 5 cos u

Pythagorean Identities sin2 u 1 cos 2 u 5 1 Cofunction Identities p sin 2 u 5 cos u 2

1

tan

cos

1 2 2 u2 5 cot u

cot

1 2 2 u2 5 csc u

csc

sec

p

1 cot u

cot u 5

1 tan u

1 1 tan2 u 5 sec 2 u

2

p

tan u 5

1 1 cot 2 u 5 csc 2 u

1 2 2 u2 5 sin u p

1 2 2 u2 5 tan u p

1 2 2 u2 5 sec u p

Even/Odd Identities sins2ud 5 2sin u

coss2ud 5 cos u

tans2ud 5 2tan u

cscs2ud 5 2csc u

secs2ud 5 sec u

cots2ud 5 2cot u

Pythagorean identities are sometimes used in radical form such as sin u 5 ± !1 2 cos 2 u or tan u 5 ± !sec 2 u 2 1 where the sign depends on the choice of u.

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 5 2 2 and tan u > 0 to find the values of all six trigonometric functions.

Solution Using a reciprocal identity, you have

Video

1 1 2 5 52 . sec u 23y2 3

cos u 5

Using a Pythagorean identity, you have

.

sin2 u 5 1 2 cos 2 u

1 32

512 2

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 5 sin x cos 2 x 2 sin x and y2 5 2sin3 x

512

2

Substitute 2 3 for cos u.

4 5 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 5 2!5y3. Now, knowing the values of the sine and cosine, you can find the values of all six trigonometric functions. sin u 5 2

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 5 2 tan u 5

3 2 3

sin u 2!5y3 !5 5 5 cos u 22y3 2

csc u 5

1 3!5 3 52 52 !5 sin u 5

sec u 5

1 3 52 cos u 2

cot u 5

1 2 2!5 5 5 ! tan u 5 5

Now try Exercise 11.

2

Example 2 −π

Simplifying a Trigonometric Expression

π

Simplify sin x cos 2 x 2 sin x. −2

Solution First factor out a common monomial factor and then use a fundamental identity.

Video .

sin x cos 2 x 2 sin x 5 sin xscos2 x 2 1d

Factor out common monomial factor.

5 2sin xs1 2 cos 2 xd

Factor out 21.

5 2sin xs

Pythagorean identity

sin2

5 2sin3 x Now try Exercise 45.

xd

Multiply.

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 u 2 1

b. 4 tan2 u 1 tan u 2 3

Solution a. Here you have the difference of two squares, which factors as sec2 u 2 1 5 ssec u 2 1dssec u 1 1). b. This expression has the polynomial form ax 2 1 bx 1 c, and it factors as 4 tan2 u 1 tan u 2 3 5 s4 tan u 2 3dstan u 1 1d. 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 2 cot x 2 3.

Solution Use the identity csc 2 x 5 1 1 cot 2 x to rewrite the expression in terms of the cotangent. csc 2 x 2 cot x 2 3 5 s1 1 cot 2 xd 2 cot x 2 3

Pythagorean identity

5 cot 2 x 2 cot x 2 2

Combine like terms.

5 scot x 2 2dscot x 1 1d

Factor.

Now try Exercise 51.

Example 5

Simplifying a Trigonometric Expression

Simplify sin t 1 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 1 cot t cos t 5 sin t 1

1 sin t 2 cos t cos t

sin2 t 1 cos 2 t sin t 1 5 sin t

5

5 csc t Now try Exercise 57.

Quotient identity

Add fractions. Pythagorean identity Reciprocal identity

Adding Trigonometric Expressions

Example 6

Perform the addition and simplify. sin u cos u 1 1 1 cos u sin u

Solution sin u cos u ssin udssin ud 1 (cos uds1 1 cos ud 1 5 1 1 cos u sin u s1 1 cos udssin ud sin2 u 1 cos2 u 1 cos u s1 1 cos udssin ud 1 1 cos u 5 s1 1 cos udssin ud 5

5

1 sin u

Multiply. Pythagorean identity: sin2 u 1 cos2 u 5 1 Divide out common factor.

5 csc u

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 1 sin x

Solution From the Pythagorean identity cos 2 x 5 1 2 sin2 x 5 s1 2 sin xds1 1 sin xd, you can see that multiplying both the numerator and the denominator by s1 2 sin xd will produce a monomial denominator. 1 1 5 1 1 sin x 1 1 sin x

1 2 sin x

? 1 2 sin x

Multiply numerator and denominator by s1 2 sin xd.

5

1 2 sin x 1 2 sin2 x

Multiply.

5

1 2 sin x cos 2 x

Pythagorean identity

5

1 sin x 2 2 cos x cos 2 x

Write as separate fractions.

5

1 sin x 2 2 cos x cos x

1

? cos x

5 sec2 x 2 tan x sec x Now try Exercise 65.

Product of fractions Reciprocal and quotient identities

Trigonometric Substitution

Example 8

Use the substitution x 5 2 tan u, 0 < u < py2, to write !4 1 x 2

as a trigonometric function of u.

Solution Begin by letting x 5 2 tan u. Then, you can obtain !4 1 x 2 5 !4 1 s2 tan ud 2

Substitute 2 tan u for x.

5 !4 1 4 tan2 u

Rule of exponents

5 !4s1 1 tan2 ud

Factor.

5 !4 sec 2 u

Pythagorean identity

5 2 sec u.

sec u > 0 for 0 < u < py2

Now try Exercise 77. Figure 1 shows the right triangle illustration of the trigonometric substitution x 5 2 tan u in Example 8. You can use this triangle to check the solution of Example 8. For 0 < u < py2, you have

2

4+

x

x

hyp 5 !4 1 x 2 .

opp 5 x, adj 5 2, and θ = arctan x 2 2 x Angle whose tangent is . 2 FIGURE 1

With these expressions, you can write the following. sec u 5 sec u 5

hyp adj !4 1 x 2

2

2 sec u 5 !4 1 x 2 So, the solution checks.

Example 9

|

Rewriting a Logarithmic Expression

|

|

|

Rewrite ln csc u 1 ln tan u as a single logarithm and simplify the result.

Solution

|

|

|

|

|

ln csc u 1 ln tan u 5 ln csc u tan u 5 ln 5 ln

|

| | | | sin u

1 sin u

? cos u

1 cos u

|

5 ln sec u

|

Now try Exercise 91.

Product Property of Logarithms Reciprocal and quotient identities

Simplify. Reciprocal identity

Exercises The symbol

indicates an exercise in which you are instructed to use graphing technology or a symbolic computer algebra system.

Click on

to view the complete solution of the exercise.

Click on

to print an enlarged copy of the graph.

Click on

to view the Make a Decision exercise.

VOCABULARY CHECK: Fill in the blank to complete the trigonometric identity.

Glossary 1.

sin u 5 ________ cos u

2.

1 5 ________ sec u

3.

1 5 ________ tan u

4.

1 5 ________ sin u

5. 1 1 ________ 5 csc2 u 7. sin

6. 1 1 tan2 u 5 ________

1p2 2 u2 5 ________

8. sec

9. coss2ud 5 ________

1p2 2 u2 5 ________

10. tans2ud 5 ________

In Exercises 1–14, use the given values to evaluate (if possible) all six trigonometric functions. 1. sin x 5 2. tan x 5

!3

2

, cos x 5 2

!3

3

,

cos x 5 2

3. sec u 5 !2, sin u 5 2 5

1 2 2 !2

2

3 7. sec f 5 , 2 8. cos

sin f 5

!10

p

3

cos x 5

4 5

10. sec x 5 4, sin x > 0 11. tan u 5 2, sin u < 0 cos u < 0

13. sin u 5 21,

cot u 5 0

14. tan u is undefined,

(a) csc x

(b) tan x

(c) sin2 x

(d) sin x tan x

(e) sec2 x

(f) sec2 x 1 tan2 x

23.

sec4

25.

sec2

x2

tan4

22. cos2 xssec2 x 2 1d x

x21 sin2 x

24. cot x sec x 26.

cos2fspy2d 2 xg cos x

In Exercises 27–44, use the fundamental identities to simplify the expression. There is more than one correct form of each answer.

!2 1 9. sins2xd 5 2 , tan x 5 2 3 4

12. csc u 5 25,

sinfspy2d 2 xg cosfspy2d 2 xg

10

3!5 csc f 5 2 5

1 2 2 x2 5 5,

20.

21. sin x sec x

tan u 5 4 5 13 5. tan x 5 12, sec x 5 2 12 6. cot f 5 23,

sins2xd coss2xd

In Exercises 21–26, match the trigonometric expression with one of the following.

!3

3

4. csc u 5 3,

19.

27. cot u sec u

28. cos b tan b

29. sin fscsc f 2 sin fd

30. sec 2 xs1 2 sin2 xd

31.

cot x csc x

32.

csc u sec u

33.

1 2 sin2 x csc2 x 2 1

34.

1 tan2 x 1 1

36.

tan2 u sec2 u

35. sec a

sin u > 0

sin a

? tan a

1 2 2 x2sec x p

In Exercises 15–20, match the trigonometric expression with one of the following.

37. cos

(a) sec x

(b) 21

(c) cot x

39.

(d) 1

(e) 2tan x

(f) sin x

41. sin b tan b 1 cos b

cos2 y 1 2 sin y

15. sec x cos x

16. tan x csc x

43. cot u sin u 1 tan u cos u

17. cot2 x 2 csc 2 x

18. s1 2 cos 2 xdscsc xd

44. sin u sec u 1 cos u csc u

38. cot

1 2 2 x2cos x p

40. cos ts1 1 tan2 td 42. csc f tan f 1 sec f

In Exercises 45–56, factor the expression and use the fundamental identities to simplify. There is more than one correct form of each answer. 2

2

2

2

2

46. sin x csc x 2 sin x

47. sin2 x sec2 x 2 sin2 x

48. cos2 x 1 cos2 x tan2 x

sec2 x 2 1 sec x 2 1

50.

cos x , 1 2 sin x

cos2 x 2 4 cos x 2 2

72. y1 5 sec4 x 2 sec2 x,

52. 1 2 2 cos2 x 1 cos4 x

73. cos x cot x 1 sin x

53. sin4 x 2 cos4 x

54. sec4 x 2 tan4 x

74. sec x csc x 2 tan x

55. csc3 x 2 csc2 x 2 csc x 1 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. ssin x 1 cos xd2 58. scot x 1 csc xdscot x 2 csc xd

76.

1 1 1 sin u cos u 1 2 cos u 1 1 sin u

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. 62.

63.

cos x 1 1 sin x 1 1 1 sin x cos x

64. tan x 2

1 1 2 sec x 1 1 sec x 2 1 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 1 sec x

3 sec x 2 tan x

1

68.

2

In Exercises 77– 82, use the trigonometric substitution to write the algebraic expression as a trigonometric function of u, where 0 < u < p/2. x 5 3 cos u

78. !64 2 16x 2,

60. s3 2 3 sin xds3 1 3 sin xd

2

1 1 2 cos x sin x cos x

77. !9 2 x 2,

59. s2 csc x 1 2ds2 csc x 2 2d

sin2 y 65. 1 2 cos y

1

75.

56. sec3 x 2 sec2 x 2 sec x 1 1

1 1 1 1 1 cos x 1 2 cos x

y2 5 tan2 x 1 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.

51. tan4 x 1 2 tan2 x 1 1

61.

1 1 sin x cos x

y2 5

2

45. tan x 2 tan x sin x

49.

71. y1 5

x 5 2 cos u

79. !x 2 2 9,

x 5 3 sec u

80. !x 2 2 4,

x 5 2 sec u

81. !x 2 1 25,

x 5 5 tan u

82. !x 2 1 100,

x 5 10 tan u

In Exercises 83– 86, use the trigonometric substitution to write the algebraic equation as a trigonometric function of u, where 2 p/2 < u < p/2. Then find sin u and cos u. 83. 3 5 !9 2 x 2, 84. 3 5 !36 2 x 2,

x 5 3 sin u x 5 6 sin u

85. 2!2 5 !16 2 4x 2,

x 5 2 cos u

86. 25!3 5 !100 2 x 2,

x 5 10 cos u

In Exercises 87–90, use a graphing utility to solve the equation for u, where 0 ≤ u < 2p.

tan2

x csc x 1 1

87. sin u 5 !1 2 cos2 u 88. cos u 5 2 !1 2 sin2 u

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

0.2

0.4

0.6

0.8

1.0

y1

1.4

90. csc u 5 !1 1 cot2 u In Exercises 91–94, rewrite the expression as a single logarithm and simplify the result.

| | | | | | | | ln|cot t| 1 lns1 1 tan2 td

91. ln cos x 2 ln sin x

y2 69. y1 5 cos

1.2

89. sec u 5 !1 1 tan2 u

92. ln sec x 1 ln sin x

1

2

p 2x , 2

70. y1 5 sec x 2 cos x,

93. y2 5 sin x y2 5 sin x tan x

94. lnscos2 td 1 lns1 1 tan2 td

In Exercises 95–98, use a calculator to demonstrate the identity for each value of u. 95. csc2 u 2 cot2 u 5 1 (a) u 5 1328, (b) u 5

2p 7

104. As x → 0 1 , cos x → j and sec x → j.

96. tan2 u 1 1 5 sec2 u

105. As x →

(a) u 5 3468, (b) u 5 3.1 97. cos

1 2 2 u2 5 sin u p

(a) u 5 808,

In Exercises 103–106, fill in the blanks. (Note: The notation x → c 1 indicates that x approaches c from the right and x → c indicates that x approaches c from the left.) p2 , sin x → j and csc x → j. 103. As x → 2

p2 , tan x → j and cot x → j. 2

106. As x → p 1 , sin x → j and csc x → j.

(b) u 5 0.8

In Exercises 107–112, determine whether or not the equation is an identity, and give a reason for your answer.

98. sins2 ud 5 2sin u 1

107. cos u 5 !1 2 sin2 u

(a) u 5 2508, (b) u 5 2 99. Friction The forces acting on an object weighing W units on an inclined plane positioned at an angle of u with the horizontal (see figure) are modeled by

mW cos u 5 W sin u where m is the coefficient of friction. Solve the equation for m and simplify the result.

108. cot u 5 !csc2 u 1 1

ssin kud 5 tan u, k is a constant. scos kud 1 5 5 sec u 110. s5 cos ud 111. sin u csc u 5 1 112. csc2 u 5 1 109.

113. Use the definitions of sine and cosine to derive the Pythagorean identity sin2 u 1 cos2 u 5 1. 114. Writing Use the Pythagorean identity sin2 u 1 cos2 u 5 1

W

to derive the other Pythagorean identities, 1 1 tan2 u 5 sec2 u and 1 1 cot2 u 5 csc2 u. 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 sxd 5 2csc x 2 sin x is given by the expression csc x cot x 2 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. s!x 1 5ds!x 2 5d

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. s2!z 1 3d

1 x 1 x15 x28 x2

118.

2x 7 2 24 x14

120.

3 6x 2 x24 42x x2

x x2 1 2 25 x 2 5

In Exercises 121–124, sketch the graph of the function. (Include two full periods.) 121. f sxd 5

1 sin p x 2

123. f sxd 5

1 p sec x 1 2 4

1

122. f sxd 5 22 tan

2

124. f sxd 5

px 2

3 cossx 2 pd 1 3 2