605
7Trigonometric Identitiesand Equations
In 1831 Michael Faraday discovered thatwhen a wire passes by a magnet, a smallelectric current is produced in the wire.Now we generate massive amounts of elec-tricity by simultaneously rotating thousandsof wires near large electromagnets. Becauseelectric current alternates its direction onelectrical wires, it is modeled accurately byeither the sine or the cosine function.
We give many examples of applicationsof the trigonometric functions to electricityand other phenomena in the examples andexercises in this chapter, including a modelof the wattage consumption of a toaster inSection 7.4, Example 5.
7.1 Fundamental Identities
7.2 Verifying Trigonometric Identities
7.3 Sum and Difference Identities
7.4 Double-Angle Identities and Half-Angle Identities
Summary Exercises on VerifyingTrigonometric Identities
7.5 Inverse Circular Functions
7.6 Trigonometric Equations
7.7 Equations Involving InverseTrigonometric Functions
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606 CHAPTER 7 Trigonometric Identities and Equations
7.1 Fundamental IdentitiesNegative-Angle Identities Fundamental Identities Using the Fundamental Identities
Figure 1
TEACHING TIP Point out that intrigonometric identities, can bean angle in degrees, a real num-ber, or a variable.
Negative-Angle Identities As suggested by the circle shown in Figure 1,an angle having the point on its terminal side has a corresponding angle
with the point on its terminal side. From the definition of sine,
(Section 5.2)
so and are negatives of each other, or
Figure 1 shows an angle in quadrant II, but the same result holds for in anyquadrant. Also, by definition,
(Section 5.2)
so
We can use these identities for and to find in terms of
Similar reasoning gives the following identities.
This group of identities is known as the negative-angle or negative-numberidentities.
Fundamental Identities In Chapter 5 we used the definitions of thetrigonometric functions to derive the reciprocal, quotient, and Pythagorean identities. Together with the negative-angle identities, these are called the fundamental identities.
Fundamental Identities
Reciprocal Identities
Quotient Identities
(continued)
tan sin cos
cot cos
sin
cot 1
tan sec
1cos
csc 1
sin
csc csc , sec sec , cot cot
tan tan .
tan sincos
sin cos
sin cos
tan :tancossin
cos cos .
cos x
rand cos
x
r,
sin sin .
sin sin
sin yr
and sin yr
,
x, yx, y
sin() = = sin yr
O x
y
yx
(x, y)
yr
(x, y)
r
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7.1 Fundamental Identities 607
TEACHING TIP Encourage studentsto memorize the identities pre-sented in this section as well assubsequent sections. Point outthat numerical values can be usedto help check whether or not anidentity was recalled correctly.
Pythagorean Identities
Negative-Angle Identities
N O T E The most commonly recognized forms of the fundamental identitiesare given above. Throughout this chapter you must also recognize alternativeforms of these identities. For example, two other forms of are
Using the Fundamental Identities One way we use these identities is tofind the values of other trigonometric functions from the value of a given trigono-metric function. Although we could find such values using a right triangle, thisis a good way to practice using the fundamental identities.
EXAMPLE 1 Finding Trigonometric Function Values Given One Value and theQuadrant
If and is in quadrant II, find each function value.
(a) (b) (c)Solution
(a) Look for an identity that relates tangent and secant.Pythagorean identity
Combine terms.
Take the negative square root. (Section 1.4)
Simplify the radical. (Section R.7)
We chose the negative square root since sec is negative in quadrant II.
34
3 sec
349 sec
349
sec2
259
1 sec2
tan 53 532 1 sec2 tan2 1 sec2
cotsin sec
tan 53
sin2 1 cos2 and cos2 1 sin2 .
sin2 cos2 1
csc() csc sec() sec cot() cot sin() sin cos() cos tan() tan
sin2 cos2 1 tan2 1 sec2 1 cot2 csc2
TEACHING TIP Warn students thatthe given information in
Example 1,
does not mean that and Ask them whythese values cannot be correct.
cos 3.sin 5
tan 53
53
,
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608 CHAPTER 7 Trigonometric Identities and Equations
(b) Quotient identity
Multiply by cos .
Reciprocal identity
(c) Reciprocal identity
Negative-angle identity
Simplify. (Section R.5)
Now try Exercises 5, 7, and 9.
C A U T I O N To avoid a common error, when taking the square root, be sure tochoose the sign based on the quadrant of and the function being evaluated.
Any trigonometric function of a number or angle can be expressed in termsof any other function.
EXAMPLE 2 Expressing One Function in Terms of Another
Express cos x in terms of tan x.
Solution Since sec x is related to both cos x and tan x by identities, start with
Take reciprocals.
Reciprocal identity
Take square roots.
Quotient rule (Section R.7); rewrite.
Rationalize the denominator.(Section R.7)
Choose the sign or the sign, depending on the quadrant of x.
Now try Exercise 43.
cos x 1 tan2 x
1 tan2 x
cos x 1
1 tan2 x
11 tan2 x cos x
11 tan2 x
cos2 x
11 tan2 x
1
sec2 x
1 tan2 x sec2 x.
tan 53 ; cot 1
53
35
cot 1
tan
cot 1
tan
sin 534
34
33434 53 sin 1
sec tan sin cos tan sin
tan sin cos
From part (a),tan 53
1sec
334
33434 ;
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7.1 Fundamental Identities 609
Figure 2 Write each fraction with theLCD. (Section R.5)
Concept Check Fill in the blanks.
1. If , then .2. If , then .3. If , then .4. If and , then .
Find sin s. See Example 1.
5. , s in quadrant I 6. , s in quadrant IV
7. , 8. ,
9. , 10.
11. Why is it unnecessary to give the quadrant of s in Exercise 10?
csc s 85tan s 0sec s
114
sec s 0tan s 7
2tan s 0coss
55
cot s 13
cos s 34
tanx sin x .6cos x .8cot x tan x 1.6
cosx cos x .65tanx tan x 2.6
1. 2.6 2. .65 3. .625
4. .75 5. 6.
7. 8.
9. 10. 58
105
11
7711
25
5
310
107
4
7.1 Exercises
9
4
4
2 2
Y1 = Y2
4
4
2 2
The graph supports the resultin Example 3. The graphs ofy1 and y2 appear to be iden-tical.
y1 = tan x + cot xy2 = cos x sin x
1
We can use a graphing calculator to decide whether two functions areidentical. See Figure 2, which supports the identity With anidentity, you should see no difference in the two graphs.
All other trigonometric functions can easily be expressed in terms of and/or We often make such substitutions in an expression to simplify it.
EXAMPLE 3 Rewriting an Expression in Terms of Sine and Cosine
Write in terms of and and then simplify theexpression.
Solution
Quotient identities
Add fractions.
Pythagorean identity
Now try Exercise 55.
C A U T I O N When working with trigonometric expressions and identities, besure to write the argument of the function. For example, we would not write
an argument such as is necessary in this identity.sin2 cos2 1;
tan cot 1
cos sin
sin2 cos2
cos sin
sin2
cos sin
cos2
cos sin
tan cot sin cos
cos
sin
cos ,sin tan cot
cos .sin
sin2 x cos2 x 1.
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610 CHAPTER 7 Trigonometric Identities and Equations
12. sin x 13. odd 14. cos x15. even 16. tan x 17. odd18.19.20.
21. ;
; ;
;
22. ; ;
; ;
23. ;
; ;
;
24. ;
; ;
;
25. ; ;
; ;
26. ; ;
; ;
27. ; ;
; ;
csc 47
7
cot 37
7tan
73
cos 34
sin 7
4
csc 54
sec 53
cot 34
tan 43
cos 35
csc 53
sec 54
tan 34
cos 45sin
35
sec 521
21cot
212
tan 221
21cos
215
sin 25
csc 17sec 17
4
cot 4cos 417
17
sin 17
17
csc 5612
sec 5cot 612
tan 26sin 26
5
csc 32
sec 35
5
cot 52
tan 25
5
cos 53
f x f xf x f xf x f x
Relating ConceptsFor individual or collaborative investigation
(Exercises 1217)A function is called an even function if for all x in the domain of f. A func-tion is called an odd function if for all x in the domain of f. WorkExercises 1217 in order, to see the connection between the negative-angle identitiesand even and odd functions.12. Complete the statement: .13. Is the function defined by even or odd?14. Complete the statement: .15. Is the function defined by even or odd?16. Complete the statement: .17. Is the function defined by even or odd?
Concept Check For each graph, determine whether or is true.
18. 19.
20.
Find the remaining five trigonometric functions of . See Example 1.
21. , in quadrant II 22. , in quadrant I
23. , in quadrant IV 24. , in quadrant III
25. , 26. ,
27. , 28. , sin 0cos 14
sin 0sec 43
cos 0sin 45sin 0cot
43
csc 52
tan 14
cos 15sin
23
4
4
2 2
4
4
2 2
4
4
2 2
f x f xf x f x
f x tan xtanx f x cos x
cosx f x sin x
sinx
f x f xf x f x
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7.1 Fundamental Identities 611
28. ;
; ;
;
29. B 30. D 31. E 32. C33. A 34. C 35. A 36. E37. D 38. B
41.
42.
43.
44.
45.46.
47.
48.
49. 50. 1 51.52. 53. 54.55.56. 57.
58. 59.
60. 61.62. tan2
cos2 cot tan
sin2 cos
sin2 cos2
cot tan 1 cot sec cos
tan2 cos2 csc2 cot cos
sec x 1 sin2 x
1 sin2 x
csc x 1 cos2 x
1 cos2 x
cot x csc2 x 1tan x sec2 x 1
cot x 1 sin2 x
sin x
sin x 1 cos2 x
tan 22p 4
p
sin 2x 1
x 1
csc 415
15sec 4
cot 1515tan 15
sin 15
4
99
Concept Check For each expression in Column I, choose the expression fromColumn II that completes an identity.
I II
29. A.
30. B. cot x31. C.
32. D.
33. E. cos x
Concept Check For each expression in Column I, choose the expression fromColumn II that completes an identity. You may have to rewrite one or both expressions.
I II
34. A.
35. B.
36. C.
37. D.38. E. tan x
39. A student writes . Comment on this students work.
40. Another student makes the following claim: Since , I should beable to also say that if I take the square root of both sides.Comment on this students statement.
41. Concept Check Suppose that . Find .
42. Concept Check Find if .
Write the first trigonometric function in terms of the second trigonometric function. SeeExample 2.
43. sin x; cos x 44. cot x; sin x 45. tan x; sec x46. cot x; csc x 47. csc x; cos x 48. sec x; sin x
Write each expression in terms of sine and cosine, and simplify it. See Example 3.49. 50. 51.52. 53. 54.
55. 56.
57. 58.
59. 60.
61. 62. 1 tan2
1 cot2 sin csc sin
sec csc cos sin sec cos
1 sin2 1 cot2
cos2 sin2 sin cos
cos sin sin
1 cos 1 sec
sec 1 sec 1sin2 csc2 1cot2 1 tan2 cos csc sec cot sin cot sin
sec p 4
ptan
sin cos xx 1
sin cos 1sin2 cos2 1
1 cot2 csc2cos2 x
csc2 x cot2 x sin2 x1 sin2 x
sinxsec x
csc x
1sec2 x
sec2 x 1
sin2 xcos2 x
tan x cos x
1
sin xcos x
tan2 x 1
sec2 xcosx tan x
sin2 x cos2 xcos x
sin x
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612 CHAPTER 7 Trigonometric Identities and Equations
63. 64.
65. ;
66. ;
67.68. It is the negative of .69.70. It is the same function.71. (a) (b)(c)72. identity 73. not an identity74. not an identity 75. identity76. not an identity
5 sin3xcos2xsin4x
cos4xsin2x
sin2x
22 89
22 89
256 6012
256 6012
sin sec2
7.2 Verifying Trigonometric IdentitiesVerifying Identities by Working with One Side Verifying Identities by Working with Both Sides
Recall that an identity is an equation that is satisfied for all meaningful replace-ments of the variable. One of the skills required for more advanced work inmathematics, especially in calculus, is the ability to use identities to write ex-pressions in alternative forms. We develop this skill by using the fundamentalidentities to verify that a trigonometric equation is an identity (for those valuesof the variable for which it is defined). Here are some hints to help you getstarted.
63. 64.
65. Concept Check Let . Find all possible values for .
66. Concept Check Let . Find all possible values for .
Relating ConceptsFor individual or collaborative investigation
(Exercises 6771)In Chapter 6 we graphed functions defined by
with the assumption that . To see what happens when , work Ex-ercises 6771 in order.
67. Use a negative-angle identity to write as a function of 2x.68. How does your answer to Exercise 67 relate to ?69. Use a negative-angle identity to write as a function of 4x.70. How does your answer to Exercise 69 relate to ?71. Use your results from Exercises 6770 to rewrite the following with a positive value
of b.(a) (b) (c)
Use a graphing calculator to decide whether each equation is an identity. (Hint: InExercise 76, graph the function of x for a few different values of y (in radians).)72. 73.74. 75.76. cosx y cos x cos y
cos 2x cos2 x sin2 xsin x 1 cos2 x
2 sin s sin 2scos 2x 1 2 sin2 x
5 sin3xcos2xsin4x
y cos4xy cos4x
y sin2xy sin2x
b 0b 0
y c a f bx d
sin x cos xsec xcsc x 3
sec x tan xsin xcos x
15
tansec
sin2 tan2 cos2
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7.2 Verifying Trigonometric Identities 613
TEACHING TIP There is no substi-tute for experience when it comesto verifying identities. Guide stu-dents through several examples,giving hints such as Apply a recip-rocal identity, or Use a differentform of the Pythagorean identity
sin2 cos2 1.
Looking Ahead to CalculusTrigonometric identities are used incalculus to simplify trigonometric ex-pressions, determine derivatives oftrigonometric functions, and changethe form of some integrals.
Hints for Verifying Identities
1. Learn the fundamental identities given in the last section. Whenever yousee either side of a fundamental identity, the other side should come tomind. Also, be aware of equivalent forms of the fundamental identities.For example, is an alternative form of the identity
2. Try to rewrite the more complicated side of the equation so that it is iden-tical to the simpler side.
3. It is sometimes helpful to express all trigonometric functions in the equa-tion in terms of sine and cosine and then simplify the result.
4. Usually, any factoring or indicated algebraic operations should be per-formed. For example, the expression can be factoredas The sum or difference of two trigonometric expressions,such as can be added or subtracted in the same way as anyother rational expression.
5. As you select substitutions, keep in mind the side you are not changing,because it represents your goal. For example, to verify the identity
try to think of an identity that relates tan x to cos x. In this case, since and the secant function is the best link
between the two sides.6. If an expression contains multiplying both numerator and de-
nominator by would give which could be replacedwith Similar results for and maybe useful.
C A U T I O N Verifying identities is not the same as solving equations.Techniques used in solving equations, such as adding the same terms to bothsides, or multiplying both sides by the same term, should not be used whenworking with identities since you are starting with a statement (to be verified)that may not be true.
Verifying Identities by Working with One Side To avoid the temptationto use algebraic properties of equations to verify identities, work with only oneside and rewrite it to match the other side, as shown in Examples 14.
1 cos x1 cos x,1 sin x,cos2 x.1 sin2 x,1 sin x
1 sin x,
sec2 x tan2 x 1,1cos xsec x
tan2 x 1 1
cos2 x,
cos sin
sin cos
1sin
1
cos
cos
sin cos
sin sin cos
1sin
1cos ,
sin x 12.sin2 x 2 sin x 1
cos2 1.sin2 sin2 1 cos2
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614 CHAPTER 7 Trigonometric Identities and Equations
4
4
2 2
The screen supports the con-clusion in Example 2.
tan2 x (1 + cot2 x) = 1 sin2 x
1
4
4
2 2
The graphs coincide, support-ing the conclusion in Exam-ple 1.
For s = x,cot x + 1 = csc x (cos x + sin x)
EXAMPLE 1 Verifying an Identity (Working with One Side)
Verify that the following equation is an identity.
Solution We use the fundamental identities from Section 7.1 to rewrite oneside of the equation so that it is identical to the other side. Since the right side ismore complicated, we work with it, using the third hint to change all functionsto sine or cosine.
Steps ReasonsRight side of
given equation
Distributive property (Section R.1)
Left side ofgiven equation
The given equation is an identity since the right side equals the left side.
Now try Exercise 33.
EXAMPLE 2 Verifying an Identity (Working with One Side)
Verify that the following equation is an identity.
Solution We work with the more complicated left side, as suggested in the sec-ond hint. Again, we use the fundamental identities from Section 7.1.
Distributive property
Since the left side is identical to the right side, the given equation is an identity.
Now try Exercise 37.
cos2 x 1 sin2 x 1
1 sin2 x
sec2 x 1cos2 x 1
cos2 x
tan2 x 1 sec2 x sec2 x
tan2 x 1tan2 x 1 tan2 x 1
cot2 x 1tan2 x tan2 x tan2 x
1tan2 x
tan2 x1 cot2 x tan2 x tan2 x cot2 x
tan2 x1 cot2 x 1
1 sin2 x
cos ssin s cot s;
sin ssin s 1 cot s 1
cos s
sin s
sin ssin s
csc s 1sin s csc scos s sin s 1
sin scos s sin s
cot s 1 csc scos s sin s
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7.2 Verifying Trigonometric Identities 615
TEACHING TIP Show that the iden-tity in Example 4 can also be veri-fied by multiplying the numeratorand denominator of the left sideby 1 sin x.
EXAMPLE 3 Verifying an Identity (Working with One Side)
Verify that the following equation is an identity.
Solution We transform the more complicated left side to match the right side.
(Section R.5)
The third hint about writing all trigonometric functions in terms of sine and cosine was used in the third line of the solution.
Now try Exercise 41.
EXAMPLE 4 Verifying an Identity (Working with One Side)
Verify that the following equation is an identity.
Solution We work on the right side, using the last hint in the list given earlierto multiply numerator and denominator on the right by
Multiply by 1. (Section R.1)
(Section R.3)
Lowest terms (Section R.5)
Now try Exercise 47.
Verifying Identities by Working with Both Sides If both sides of anidentity appear to be equally complex, the identity can be verified by workingindependently on the left side and on the right side, until each side is changedinto some common third result. Each step, on each side, must be reversible.
cos x
1 sin x
1 sin2 x cos2 x cos2 x
cos x1 sin x
x y x y x2 y2 1 sin2 x
cos x1 sin x
1 sin xcos x
1 sin x 1 sin x
cos x1 sin x
1 sin x.
cos x
1 sin x
1 sin xcos x
1cos2 t sec
2 t; 1sin2 t csc
2 t sec2 t csc2 t
1
cos2 t
1sin2 t
tan t sin tcos t ; cot t cos tsin t
sin tcos t
1
sin t cos t
cos t
sin t
1sin t cos t
ab a
1b tan t
1sin t cos t
cot t 1
sin t cos t
a bc
ac
bc
tan t cot tsin t cos t
tan t
sin t cos t
cot tsin t cos t
tan t cot tsin t cos t
sec2 t csc2 t
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616 CHAPTER 7 Trigonometric Identities and Equations
(Section R.4)x2 2xy y2 x y2
With all steps reversible, the procedure is as shown in the margin. The left sideleads to a common third expression, which leads back to the right side. This pro-cedure is just a shortcut for the procedure used in Examples 14: one side ischanged into the other side, but by going through an intermediate step.
EXAMPLE 5 Verifying an Identity (Working with Both Sides)
Verify that the following equation is an identity.
Solution Both sides appear equally complex, so we verify the identity bychanging each side into a common third expression. We work first on the left,multiplying numerator and denominator by cos .
Multiply by 1.
Distributive property
On the right side of the original equation, begin by factoring.
Factor
Lowest terms
We have shown that
Left side of Common third Right side ofgiven equation expression given equation
verifying that the given equation is an identity.
Now try Exercise 51.
sec tan sec tan
1 sin 1 sin
1 2 sin sin2
cos2 ,
1 sin 1 sin
1 sin2 . 1 sin 2
1 sin 1 sin
cos2 1 sin2 1 sin 2
1 sin2
1 2 sin sin2 cos2
1 sin 2
cos2
1 sin 1 sin
tan sin cos
1 sin cos
cos
1 sin cos
cos
sec cos 1 1 tan cos 1 tan cos
sec cos tan cos sec cos tan cos
sec tan sec tan
sec tan cos sec tan cos
sec tan sec tan
1 2 sin sin2
cos2
left right
common thirdexpression
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7.2 Verifying Trigonometric Identities 617
CL
An Inductor and a Capacitor
Figure 3
C A U T I O N Use the method of Example 5 only if the steps are reversible.
There are usually several ways to verify a given identity. For instance, an-other way to begin verifying the identity in Example 5 is to work on the left asfollows.
Fundamental identities (Section 7.1)
Compare this with the result shown in Example 5 for the right side to see thatthe two sides indeed agree.
EXAMPLE 6 Applying a Pythagorean Identity to Radios
Tuners in radios select a radio station by adjusting the frequency. A tuner maycontain an inductor L and a capacitor C, as illustrated in Figure 3. The energystored in the inductor at time t is given by
and the energy stored in the capacitor is given by
where F is the frequency of the radio station and k is a constant. The total energyE in the circuit is given by
Show that E is a constant function. (Source: Weidner, R. and R. Sells, Ele-mentary Classical Physics, Vol. 2, Allyn & Bacon, 1973.)Solution
Given equation
Substitute.
Factor. (Section R.4)
Since k is constant, is a constant function.
Now try Exercise 85.
Et
ksin2 cos2 1 (Here 2Ft). k1
ksin22Ft cos22Ft k sin22Ft k cos22Ft
Et Lt Ct
Et Lt Ct.
Ct k cos22Ft,
Lt k sin22Ft
1 sin 1 sin
1 sin cos
1 sin cos
sec tan sec tan
1cos
sin cos
1cos
sin cos
Add and subtract fractions.(Section R.5)
Simplify the complex fraction.(Section R.5)
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618 CHAPTER 7 Trigonometric Identities and Equations
Perform each indicated operation and simplify the result.
1. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11. 12.
Factor each trigonometric expression.
13. 14.15. 16.17. 18.19. 20.21. 22.
Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.23. 24. 25. sec r cos r
26. cot t tan t 27. 28.
29. 30. 31.
32.
In Exercises 3368, verify that each trigonometric equation is an identity. SeeExamples 15.
33. 34.
35. 36.
37. 38.39. 40.
41. 42.
43. 44.
45.46. tan2 sin2 tan2 cos2 1
1 cos2 1 cos2 2 sin2 sin4
cos
sin cot 1sin4 cos4 2 sin2 1
sin2 cos
sec cos cos
sec
sin csc
sec2 tan2
sin2 tan2 cos2 sec2 cot s tan s sec s csc ssin2 1 cot2 1cos2 tan2 1 1
tan2 1sec
sec 1 sin2
cos cos
tan sec
sin cot csc
cos
1tan2
cot tan
sin2 xcos2 x
sin x csc xcsc2 t 1sec2 x 1
csc sec
cot sin tan
cos
cot sin tan cos
sin3 cos3 sin3 x cos3 xcot4 x 3 cot2 x 2cos4 x 2 cos2 x 14 tan2 tan 32 sin2 x 3 sin x 1tan x cot x2 tan x cot x2sin x 12 sin x 12sec2 1sin2 1
sin cos 21
1 cos x
11 cos x
1 tan s2 2 tan s
1 sin t2 cos2 tcos
sin
sin 1 cos
cos x
sec x
sin xcsc x
1sin 1
1
sin 11
csc2
1sec2
cos sec csc
tan scot s csc ssec x
csc x
csc x
sec xcot
1cot
1. or
2. csc x sec x or
3. 4.
5. 1 6. or
7. 1 8. or
9. 10.
11. or
12.13.14.15. 4 sin x 16. 4 tan x cot x or 417.18.19.20. or
21.22.23. 24. 25. 126. 1 27. 28.29. 30. 31.32. csc2
sec2 xcot2 ttan2 xsec2 tan2
cos sin sin cos 1 sin cos sin x cos x 1 sin x cos x
csc2 xcot2 x 2cot2 x 2 cot2 x 1cos2 x 124 tan 3 tan 12 sin x 1 sin x 1
sec 1 sec 1sin 1 sin 11 2 sin cos
2 cot x csc x2 cos xsin2 x
sec2 s2 2 sin t
csc 1
sin
2 sec2 2
cos2
1 cot 1 sec s
1sin x cos x
1sin cos
csc sec
7.2 Exercises
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7.2 Verifying Trigonometric Identities 619
47. 48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69. A student claims that the equation is an identity, since by lettingor radians we get , a true statement. Comment on this stu-
dents reasoning.70. An equation that is an identity has an infinite number of solutions. If an equation has
an infinite number of solutions, is it necessarily an identity? Explain.
0 1 12 90cos sin 1
sin4 cos4 sin2 cos2
1
sec csc cos sin cot tan
sec tan 2 1 sin 1 sin
1 cos x1 cos x
1 cos x1 cos x
4 cot x csc x
1 sin x cos x2 21 sin x 1 cos x
tan2 t 1sec2 t
tan t cot ttan t cot t
cot2 t 11 cot2 t
1 2 sin2 t
sec4 s tan4 ssec2 s tan2 s
sec2 s tan2 s
sin 1 cos
sin cos 1 cos
csc 1 cos2
sin cos sin
1 cos
sin
cos
1 sin cos
1 sin 1 sin
sec2 2 sec tan tan2
sec4 x sec2 x tan4 x tan2 x
csc cot tan sin
cot csc
sin2 sec2 sin2 csc2 sec2
1tan sec
1
tan sec 2 tan
cot 1cot 1
1 tan 1 tan
1 cos x1 cos x
cot x csc x2
tan s1 cos s
sin s
1 cos s cot s sec s csc s
1sec tan
sec tan
11 sin
1
1 sin 2 sec2
sec tan 2 1sec csc tan csc
2 tan cos 1
tan2
cos
sec 1
9
9
LIALMC07_0321227638.QXP 2/26/04 10:47 AM Page 619
620 CHAPTER 7 Trigonometric Identities and Equations
Concept Check Graph each expression and conjecture an identity. Then verify yourconjecture algebraically.71. 72.
73. 74.
Graph the expressions on each side of the equals sign to determine whether the equationmight be an identity. (Note: Use a domain whose length is at least 2 .) If the equationlooks like an identity, prove it algebraically. See Example 1.
75. 76.
77. 78.
By substituting a number for s or t, show that the equation is not an identity.79. 80.81. 82.
83. When is a true statement?
(Modeling) Work each problem.84. Intensity of a Lamp According to Lamberts law, the
intensity of light from a single source on a flat surface atpoint P is given by
,
where k is a constant. (Source: Winter, C., Solar PowerPlants, Springer-Verlag, 1991.)(a) Write I in terms of the sine function.(b) Explain why the maximum value of I occurs when
.
85. Oscillating Spring The distance or displacement y of aweight attached to an oscillating spring from its natural posi-tion is modeled by
,
where t is time in seconds. Potential energy is the energy ofposition and is given by
,
where k is a constant. The weight has the greatest potential energy when the springis stretched the most. (Source: Weidner, R. and R. Sells, Elementary ClassicalPhysics, Vol. 1, Allyn & Bacon, 1973.)(a) Write an expression for P that involves the cosine function.(b) Use a fundamental identity to write P in terms of .
86. Radio Tuners Refer to Example 6. Let the energy stored in the inductor be given by
and the energy in the capacitor be given by
,Ct 3 sin26,000,000t
Lt 3 cos26,000,000t
sin2t
P ky2
y 4 cos2t
0
I k cos2
sin x 1 cos2 x
cos t 1 sin2 tcsc t 1 cot2 tcos2 s cos ssincsc s 1
11 sin s
1
1 sin s sec2 s
tan s cot stan s cot s
2 sin2 s
1 cot2 s sec2 s
sec2 s 12 5 cos s
sin s 2 csc s 5 cot s
tan sin cos cos 1
sin tan
csc cot sec 1sec tan 1 sin
71.
72.
73.
74.75. identity 76. identity77. not an identity78. not an identity83. It is true when 84. (a)(b) For for all integers n,
, its maximum value,and I attains a maximum value of k.85. (a)(b) P 16k1 sin22t
P 16k cos22t
cos2 1 2nI k1 sin2
sin x 0.
tan sin cos sec
cos 1sin tan
cot
tan csc cot sec 1
cos sec tan 1 sin
9 P
x
y
LIALMC07_0321227638.QXP 2/26/04 10:47 AM Page 620
where t is time in seconds. The total energy E in the circuit is given by.
(a) Graph L, C, and E in the window by , with and. Interpret the graph.
(b) Make a table of values for L, C, and E starting at , incrementing by .Interpret your results.
(c) Use a fundamental identity to derive a simplified expression for .Et
107t 0Yscl 1
Xscl 1071, 40, 106Et Lt Ct
7.3 Sum and Difference Identities 621
86. (a) The sum of L and Cequals 3.
4
0 106
1
7.3 Sum and Difference IdentitiesCosine Sum and Difference Identities Cofunction Identities Sine and Tangent Sum and Difference Identities
Cosine Sum and Difference Identities Several examples presented ear-lier should have convinced you by now that does not equal
For example, if and then
while
We can now derive a formula for We start by locating angles Aand B in standard position on a unit circle, with Let S and Q be thepoints where the terminal sides of angles A and B, respectively, intersect thecircle. Locate point R on the unit circle so that angle POR equals the difference
See Figure 4.
Figure 4
Point Q is on the unit circle, so by the work with circular functions inChapter 6, the x-coordinate of Q is the cosine of angle B, while the y-coordinateof Q is the sine of angle B.
Q has coordinates In the same way,
S has coordinates
and R has coordinates cosA B, sinA B.
cos A, sin A,
cos B, sin B.
Ox
y
(cos(A B), sin(A B)) R
BA A B
(cos A, sin A) SP (1, 0)
Q (cos B, sin B
A B.
B A.cosA B.
cos A cos B cos
2 cos 0 0 1 1.
cosA B cos 2 0 cos 2 0,B 0,A 2cos A cos B.
cosA B
(b) Let , , and . for all inputs.(c) Et 3
Y3 3Y3 EtY2 CtY1 Lt
LIALMC07_0321227638.QXP 2/26/04 10:47 AM Page 621
622 CHAPTER 7 Trigonometric Identities and Equations
Angle SOQ also equals Since the central angles SOQ and POR areequal, chords PR and SQ are equal. By the distance formula, since
(Section 2.1)
Squaring both sides and clearing parentheses gives
Since for any value of x, we can rewrite the equation as
Subtract 2; divide by 2.
Although Figure 4 shows angles A and B in the second and first quadrants,respectively, this result is the same for any values of these angles.
To find a similar expression for rewrite as and use the identity for
Cosine difference identity
Negative-angle identities (Section 7.1)
Cosine of a Sum or Difference
These identities are important in calculus and useful in certain applications.Although a calculator can be used to find an approximation for cos 15, for example, the method shown below can be applied to get an exact value, as wellas to practice using the sum and difference identities.
EXAMPLE 1 Finding Exact Cosine Function Values
Find the exact value of each expression.
(a) cos 15 (b) (c)
Solution
(a) To find cos 15, we write 15 as the sum or difference of two angles withknown function values. Since we know the exact trigonometric functionvalues of 45 and 30, we write 15 as (We could also use
) Then we use the identity for the cosine of the difference of two angles.60 45.
45 30.
cos 87 cos 93 sin 87 sin 93cos 5
12
cos(A B) cos A cos B sin A sin Bcos(A B) cos A cos B sin A sin B
cosA B cos A cos B sin A sin B
cos A cos B sin Asin B cos A cosB sin A sinB
cosA B cosA B
cosA B.A BA BcosA B,
cosA B cos A cos B sin A sin B. 2 2 cosA B 2 2 cos A cos B 2 sin A sin B
sin2 x cos2 x 1
cos2 A 2 cos A cos B cos2 B sin2 A 2 sin A sin B sin2 B.cos2A B 2 cosA B 1 sin2A B
cos A cos B2 sin A sin B2.
cosA B 12 sinA B 02PR SQ,
A B.
LIALMC07_0321227638.QXP 2/26/04 10:47 AM Page 622
7.3 Sum and Difference Identities 623
TEACHING TIP In Example 1(b),students may benefit from con-
verting radians to 75 in
order to realize that
can be used in place of 5
12
.
6
4 30 45
5
12
TEACHING TIP Mention that theseidentities state that the trigono-metric function of an acute angleis the same as the cofunction of itscomplement. Verify the cofunctionidentities for acute angles usingcomplementary angles in a righttriangle along with the right-triangle-based definitions of thetrigonometric functions. Emphasizethat these identities apply to anyangle , not just acute angles.
The screen supports the solu-tion in Example 1(b) byshowing that
cos
= .512
6 24
Cosine difference identity
(Section 5.3)
(b)
Cosine sum identity
(Section 6.2)
(c)Cosine sum identity
(Section 5.2)
Now try Exercises 7, 9, and 11.
Cofunction Identities We can use the identity for the cosine of the differ-ence of two angles and the fundamental identities to derive cofunctionidentities.
Cofunction Identities
Similar identities can be obtained for a real number domain by replacing90 with
Substituting 90 for A and for B in the identity for gives
This result is true for any value of since the identity for is true forany values of A and B.
cosA B sin . 0 cos 1 sin
cos90 cos 90 cos sin 90 sin cosA B
2 .
tan(90 ) cot csc(90 ) sec sin(90 ) cos sec(90 ) csc cos(90 ) sin cot(90 ) tan
1 cos 180
cos 87 cos 93 sin 87 sin 93 cos87 93
6 2
4
32
22
12
22
cos
6 cos
4 sin
6 sin
4
6
2
12 ;
4
3
12 cos
5
12
cos 6 4
6 24
22
32
22
12
cos 45 cos 30 sin 45 sin 30 cos 15 cos45 30
LIALMC07_0321227638.QXP 2/26/04 10:47 AM Page 623
624 CHAPTER 7 Trigonometric Identities and Equations
TEACHING TIP Verify results fromExample 2 using a calculator.
EXAMPLE 2 Using Cofunction Identities to Find
Find an angle that satisfies each of the following.
(a) (b) (c)
Solution
(a) Since tangent and cotangent are cofunctions,
Cofunction identity
Set angle measures equal.
(b)Cofunction identity
(c)
Cofunction identity
Combine terms.
Now try Exercises 25 and 27.
N O T E Because trigonometric (circular) functions are periodic, the solutionsin Example 2 are not unique. We give only one of infinitely many possibilities.
If one of the angles A or B in the identities for and is a quadrantal angle, then the identity allows us to write the expression in termsof a single function of A or B.
EXAMPLE 3 Reducing to a Function of a Single Variable
Write as a trigonometric function of .
Solution Use the difference identity. Replace A with 180 and B with .
(Section 5.2)
Now try Exercise 39.
cos
1 cos 0 sin cos180 cos 180 cos sin 180 sin
cos180
cosA B
cosA BcosA B
4
sec 4 sec sec 2 34 sec
csc 3
4
sec
120 90 30
cos90 cos30 sin cos30
65 90 25
tan90 tan 25 cot tan 25
tan90 cot .
csc 3
4
sec sin cos30cot tan 25
LIALMC07_0321227638.QXP 2/26/04 10:47 AM Page 624
Sine and Tangent Sum and Difference Identities We can use the cosine sum and difference identities to derive similar identities for sine and tangent. Since we replace with to get
Cofunction identity
Cosine difference identity
Cofunction identities
Now we write as and use the identity for
Sine sum identity
Negative-angle identities
Sine of a Sum or Difference
To derive the identity for we start with
We express this result in terms of the tangent function by multiplying both nu-merator and denominator by
Replacing B with and using the fact that gives theidentity for the tangent of the difference of two angles.
tanB tan BB
tan sin cos tanA B tan A tan B
1 tan A tan B
sin Acos A
sin Bcos B
1 sin Acos A
sin Bcos B
sin A cos Bcos A cos B
cos A sin Bcos A cos B
cos A cos Bcos A cos B
sin A sin Bcos A cos B
tanA B
sin A cos B cos A sin B1
cos A cos B sin A sin B1
1cos A cos B
1cos A cos B
1cos A cos B .
sin A cos B cos A sin Bcos A cos B sin A sin B
.
tanA B sinA BcosA B
tanA B,
sin(A B) sin A cos B cos A sin Bsin(A B) sin A cos B cos A sin B
sinA B sin A cos B cos A sin B sin A cosB cos A sinB
sinA B sinA B
sinA B.sinA BsinA B
sinA B sin A cos B cos A sin B.
cos90 A cos B sin90 A sin B cos90 A B
sinA B cos90 A B
A Bsin cos90 ,
7.3 Sum and Difference Identities 625
Fundamental identity(Section 7.1)
Sum identities
Simplify thecomplex fraction.(Section R.5)
Multiply numerators;multiply denominators.
LIALMC07_0321227638.QXP 2/26/04 10:47 AM Page 625
626 CHAPTER 7 Trigonometric Identities and Equations
Tangent of a Sum or Difference
EXAMPLE 4 Finding Exact Sine and Tangent Function Values
Find the exact value of each expression.
(a) sin 75 (b) (c)
Solution
(a)Sine sum identity
(Section 5.3)
(b)
Tangent sum identity
(Section 6.2)
Rationalize the denominator. (Section R.7)
Multiply. (Section R.7)
Combine terms.
Factor out 2. (Section R.5)
Lowest terms
(c)Sine difference identity
Negative-angle identity
(Section 5.3)
Now try Exercises 29, 31, and 35.
32
sin 120 sin120
sin 40 cos 160 cos 40 sin 160 sin40 160
2 3
22 3
2
4 23
2
3 3 1 3
1 3
3 11 3
1 31 3
3 1
1 3 1
tan 3 tan
4
1 tan 3 tan
4
tan 7
12
tan 3 4
6 24
22
32
22
12
sin 45 cos 30 cos 45 sin 30 sin 75 sin45 30
sin 40 cos 160 cos 40 sin 160tan 7
12
tan(A B) tan A tan B1 tan A tan B
tan(A B) tan A tan B1 tan A tan B
LIALMC07_0321227638.QXP 2/26/04 10:47 AM Page 626
7.3 Sum and Difference Identities 627
EXAMPLE 5 Writing Functions as Expressions Involving Functions of
Write each function as an expression involving functions of .
(a) (b) (c)Solution
(a) Using the identity for
(b)
(c)
Now try Exercises 43 and 47.
EXAMPLE 6 Finding Function Values and the Quadrant of
Suppose that A and B are angles in standard position, with and Find each of the following.
(a) (b) (c) the quadrant of Solution
(a) The identity for requires sin A, cos A, sin B, and cos B. We aregiven values of sin A and cos B. We must find values of cos A and sin B.
Fundamental identity
Subtract
Since A is in quadrant II,
In the same way, Now use the formula for
(b) To find first use the values of sine and cosine from part (a) toget and
tanA B
43
125
1 43 125
1615
1 4815
16156315
1663
tan B 125 .tan A 43
tanA B,
2065
3665
1665
sinA B 45 513 351213
sinA B.sin B 1213 .
cos A 0. cos A 35
1625 . cos
2 A
925
sin A 45 1625 cos
2 A 1
sin2 A cos2 A 1
sinA B
A BtanA BsinA B
B 32 .cos B 5
13 ,
2 A ,
sin A 45 ,
A B
sin 0 cos 1 sin
sin180 sin 180 cos cos 180 sin
tan45 tan 45 tan 1 tan 45 tan
1 tan 1 tan
12
cos 32
sin .
sin30 sin 30 cos cos 30 sin sinA B,
sin180 tan45 sin30
LIALMC07_0321227638.QXP 2/26/04 10:47 AM Page 627
628 CHAPTER 7 Trigonometric Identities and Equations
(c) From parts (a) and (b), and both posi-tive. Therefore, must be in quadrant I, since it is the only quadrant inwhich both sine and tangent are positive.
Now try Exercise 51.
EXAMPLE 7 Applying the Cosine Difference Identity to Voltage
Common household electric current is called alternating current because thecurrent alternates direction within the wires. The voltage V in a typical 115-voltoutlet can be expressed by the function where is the angu-lar speed (in radians per second) of the rotating generator at the electrical plantand t is time measured in seconds. (Source: Bell, D., Fundamentals of ElectricCircuits, Fourth Edition, Prentice-Hall, 1988.)(a) It is essential for electric generators to rotate at precisely 60 cycles per sec
so household appliances and computers will function properly. Determine for these electric generators.
(b) Graph V in the window by (c) Determine a value of so that the graph of is the
same as the graph of
Solution
(a) Each cycle is 2 radians at 60 cycles per sec, so the angular speed isper sec.
(b) Because the amplitude of the functionis 163 (from Section 6.3), is an appropriate interval for therange, as shown in Figure 5.
Figure 5
(c) Using the negative-angle identity for cosine and a cofunction identity,
Therefore, if then
Now try Exercise 81.
Vt 163 cost 2 163 sin t. 2 ,
cosx 2 cos 2 x cos 2 x sin x.
200
0 .05
200
For x = t,V(t) = 163 sin 120t
200, 200Vt 163 sin t 163 sin 120 t.
120 radians 602
Vt 163 sin t.Vt 163 cost
200, 200.0, .05
Vt 163 sin t,
A BtanA B 1663,sinA B
1665
LIALMC07_0321227638.QXP 2/26/04 10:47 AM Page 628
Concept Check Match each expression in Column I with the correct expression inColumn II to form an identity.
I II1. A.2. B.3. C.4. D.
E.F.
Use identities to find each exact value. (Do not use a calculator.) See Example 1.5. cos 75 6.7. cos 105 8.
(Hint: ) (Hint: )
9. 10.
11. 12.
Write each function value in terms of the cofunction of a complementary angle. SeeExample 2.
13. tan 87 14. sin 15 15. 16.
17. 18. 19. 20.
Use the cofunction identities to fill in each blank with the appropriate trigonometricfunction name. See Example 2.
21. 22.
23. 24.
Find an angle that makes each statement true. See Example 2.
25. 26.27. 28.
Use identities to find the exact value of each of the following. See Example 4.
29. 30. 31.
32. 33. 34.
35. 36.
37. 38. tan 80 tan551 tan 80 tan55
tan 80 tan 551 tan 80 tan 55
sin 40 cos 50 cos 40 sin 50sin 76 cos 31 cos 76 sin 31
tan712sin712sin 12tan
12tan
5
12
sin 5
12
cot 10 tan2 20sin3 15 cos 25sin cos2 10tan cot45 2
72 cot 1833 sin 57
6 sin 23 6cot 3
tan 174 3sec 146 42cot 9
10
sin 5
8
sin 2
5cos
12
cos 7
9
cos 2
9
sin 7
9
sin 2
9
cos 40 cos 50 sin 40 sin 50
cos 12cos 712105 60 45105 60 45
cos105cos15
cos x cos y sin x sin ycos x sin y sin x cos ysin x cos y cos x sin ysinx y sin x cos y cos x sin ysinx y sin x sin y cos x cos ycosx y cos x cos y sin x sin ycosx y
7.3 Sum and Difference Identities 629
1. F 2. A 3. C 4. D
5. 6.
7. 8.
9. 10.
11. 0 12. 1 13. cot 3
14. cos 75 15.
16. 17.
18. 19.
20. 21. tan22. cos 23. cos 24. tan
25. 15 26. 27. 20
28. 29.
30. 31.
32. 33.
34. 35. 36. 1
37. 1 38. 1
22
2 3
6 24
6 24
2 32 3
6 24
803
1003
cot84 3
csc56 42tan25 cos 8 cos 10sin
5
12
6 24
2 64
2 64
2 64
6 24
6 24
7.3 Exercises
LIALMC07_0321227638.QXP 2/26/04 10:47 AM Page 629
630 CHAPTER 7 Trigonometric Identities and Equations
39. 40.41. sin x 42. sin x
43.
44.
45.
46.
47.
48.
49. 50.
51. (a) (b) (c) (d) I
52. (a) (b)
(c) (d) IV
53. (a)
(b)
(c) (d) II
54. (a) (b) (c) (d) IV
55. (a) (b) (c)(d) II56. (a) (b) (c) (d) III
58.
59.
60. (a)
(b)
61. 62.
63. 64.
65. 66. 6 24
2 3
2 36 2
4
2 36 2
4
6 24
2 64
6 24
6 24
3677
8485
7785
7736
1385
3685
6316
3365
1665
85 5220 210
42 59
210 29
86 34 66
86 325
4 6625
6316
3365
1665
1 tan x1 tan x
3 tan x 13 tan x
3 tan 13 tan
3 tan 1 3 tan
22
cos x sin x
12 cos x 3 sin xsin
22
cos sin
cos sin Use identities to write each expression as a function of x or . See Examples 3 and 5.
39. 40. 41.
42. 43. 44.
45. 46. 47.
48. 49. 50.
Use the given information to find (a) , (b) , (c) , and (d) thequadrant of . See Example 6.
51. and , s and t in quadrant I
52. and , s and t in quadrant II
53. and , s in quadrant II and t in quadrant IV
54. and , s in quadrant I and t in quadrant III
55. and , s and t in quadrant III
56. and , s in quadrant II and t in quadrant I
Relating ConceptsFor individual or collaborative investigation
(Exercises 5760)The identities for and can be used to find exact values of expres-sions like cos 195 and cos 255, where the angle is not in the first quadrant. WorkExercises 5760 in order, to see how this is done.
57. By writing 195 as , use the identity for to express cos 195as cos 15.
58. Use the identity for to find cos 15.59. By the results of Exercises 57 and 58, .60. Find each exact value using the method shown in Exercises 5759.
(a) cos 255 (b)
Find each exact value. Use the technique developed in Relating Concepts Exer-cises 5760.
61. sin 165 62. tan 165 63. sin 255
64. tan 285 65. 66. sin1312 tan 1112
cos 11
12
cos 195 cosA B
cosA B180 15
cosA BcosA B
sin t 45cos s
1517
cos t 35cos s
817
sin t 1213
sin s 35
sin t 13
sin s 23
sin t 35cos s
15
sin t 513
cos s 35
s ttans tsins tcoss t
tan 4 xtanx 6 tan 30tan60 sin 4 xsin56 xsin180 sin45 cos 2 xcosx 32 cos180 cos90
LIALMC07_0321227638.QXP 2/26/04 10:47 AM Page 630
7.3 Sum and Difference Identities 631
71.
72.
79. 3 80. 163 and 163; no81. (a) 425 lb (c) 0
1 tan x1 tan x
tan 4 xsin 2 x cos x
9
9
67. Use the identity , and replace with , to derive theidentity .
68. Explain how the identities for , , and can be foundby using the sum identities given in this section.
69. Why is it not possible to use a method similar to that of Example 5(c) to find a for-mula for ?
70. Concept Check Show that if A, B, and C are angles of a triangle, then.
Graph each expression and use the graph to conjecture an identity. Then verify yourconjecture algebraically.
71. 72.
Verify that each equation is an identity.73.
74.
75.
76.
77.
78.
Exercises 79 and 80 refer to Example 7.79. How many times does the current oscillate in .05 sec?80. What are the maximum and minimum voltages in this outlet? Is the voltage always
equal to 115 volts?
(Modeling) Solve each problem.81. Back Stress If a person bends at the
waist with a straight back making anangle of degrees with the horizontal,then the force F exerted on the backmuscles can be modeled by the equation
,
where W is the weight of the person.(Source: Metcalf, H., Topics inClassical Biophysics, Prentice-Hall,1980.)(a) Calculate F when lb and .(b) Use an identity to show that F is approximately equal to .(c) For what value of is F maximum?
2.9W cos 30W 170
F .6W sin 90
sin 12
sins tsin t
coss t
cos t
sin ssin t cos t
sinx ysinx y
tan x tan ytan x tan y
sins tcos s cos t
tan s tan t
cos cos sin
tan cot
tanx y tan y x 2tan x tan y1 tan x tan y
sinx y sinx y 2 sin x cos y
1 tan x1 tan x
sin 2 x
sinA B C 0
tan270
cotA BcscA BsecA Bcos A sin90 A
90 Acos90 sin
LIALMC07_0321227638.QXP 2/26/04 10:47 AM Page 631
7.4 Double-Angle Identities and Half-Angle IdentitiesDouble-Angle Identities Product-to-Sum and Sum-to-Product Identities Half-Angle Identities
Double-Angle Identities When in the identities for the sum of twoangles, these identities are called the double-angle identities. For example, toderive an expression for cos 2A, we let in the identity
Cosine sum identity (Section 7.3)
cos 2A cos2 A sin2 A cos A cos A sin A sin A
cos 2A cosA A
cos A cos B sin A sin B.cosA B B A
A B
TEACHING TIP A common error isto write as 2 cos A.cos 2A
60
0 .05
60
For x = t, V = V1 + V2 = 30 sin 120t + 40 cos 120t
82. Back Stress Refer to Exercise 81.(a) Suppose a 200-lb person bends at the waist so that . Estimate the force
exerted on the persons back muscles.(b) Approximate graphically the value of that results in the back muscles exerting
a force of 400 lb.83. Sound Waves Sound is a result of waves applying pressure to a persons eardrum.
For a pure sound wave radiating outward in a spherical shape, the trigonometricfunction defined by
can be used to model the sound pressure at a radius of r feet from the source, wheret is time in seconds, is length of the sound wave in feet, c is speed of sound in feetper second, and a is maximum sound pressure at the source measured in poundsper square foot. (Source: Beranek, L., Noise and Vibration Control, Institute of Noise Control Engineering, Washington, D.C., 1988.) Let ft and
ft per sec.(a) Let lb per . Graph the sound pressure at distance ft from its
source in the window by Describe P at this distance.(b) Now let and . Graph the sound pressure in the window by
What happens to pressure P as radius r increases?(c) Suppose a person stands at a radius r so that , where n is a positive
integer. Use the difference identity for cosine to simplify P in this situation.84. Voltage of a Circuit When the two voltages
and
are applied to the same circuit, the resulting voltage V will be equal to their sum.(Source: Bell, D., Fundamentals of Electric Circuits, Second Edition, RestonPublishing Company, 1981.)(a) Graph the sum in the window by (b) Use the graph to estimate values for a and so that .(c) Use identities to verify that your expression for V is valid.
V a sin120t 60, 60.0, .05
V2 40 cos 120 tV1 30 sin 120t
r n2, 2.
0, 20]t 10a 3.05, .05.0, .05
r 10ft2a .4c 1026
4.9
P a
r cos2r ct
45
632 CHAPTER 7 Trigonometric Identities and Equations
82. (a) 408 lb (b) 46.183. (a) The pressure P isoscillating.
(b) The pressure oscillates andamplitude decreases as rincreases.
(c)84. (a)
(b) ; 5.353a 50
P a
n cos ct
0 20
2
2
For x = r,P(r) = cos[ 10,260]3r 2r4.9
0 .05
.05
.05
For x = t,P(t) = cos[ 1026t].410 204.9
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7.4 Double-Angle Identities and Half-Angle Identities 633
TEACHING TIP Students might findit helpful to see each formula illus-trated with a concrete examplethat they can check. For instance,you might show that
cos2 30 sin2 30.
cos 60 cos2 30
Looking Ahead to CalculusThe identities
and
can be rewritten as
and
These identities are used to integratethe functions and
cos2 A.gA f A sin2 A
cos2 A 12
1 cos 2A.
sin2 A 12
1 cos 2 A
cos 2A 2 cos2 A 1
cos 2A 1 2 sin2 A
Two other useful forms of this identity can be obtained by substitutingeither or Replace with the expression to get
Fundamental identity (Section 7.1)
or replace with to get
Fundamental identity
We find sin 2A with the identity letting
Sine sum identity
Using the identity for we find tan 2A.
Tangent sum identity
Double-Angle Identities
EXAMPLE 1 Finding Function Values of 2 Given Information about
Given and find and
Solution To find we must first find the value of sin .
Simplify.
Choose the negative square root since sin 0. sin 45
sin2 1625
cos 35sin2 cos2 1; sin2 352 1sin 2,
tan 2.cos 2,sin 2,sin 0,cos 35
tan 2A 2 tan A
1 tan2 A
cos 2A 2 cos2 A 1 sin 2A 2 sin A cos A cos 2A cos2 A sin2 A cos 2A 1 2 sin2 A
tan 2A 2 tan A
1 tan2 A
tan A tan A
1 tan A tan A
tan 2A tanA A
tanA B,
sin 2A 2 sin A cos A sin A cos A cos A sin A
sin 2A sinA A
B A.sinA B sin A cos B cos A sin B,
cos 2A 2 cos2 A 1. cos2 A 1 cos2 A cos2 A 1 cos2 A
cos 2A cos2 A sin2 A
1 cos2 Asin2 A
cos 2A 1 2 sin2 A, 1 sin2 A sin2 A
cos 2A cos2 A sin2 A1 sin2 A
cos2 Asin2 A 1 cos2 A.cos2 A 1 sin2 A
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634 CHAPTER 7 Trigonometric Identities and Equations
Using the double-angle identity for sine,
Now we find using the first of the double-angle identities for cosine.(Any of the three forms may be used.)
The value of can be found in either of two ways. We can use the double-
angle identity and the fact that
Simplify. (Section R.5)
Alternatively, we can find by finding the quotient of and
Now try Exercise 9.
EXAMPLE 2 Verifying a Double-Angle Identity
Verify that the following equation is an identity.
Solution We start by working on the left side, using the hint from Section 7.1about writing all functions in terms of sine and cosine.
Quotient identity
Double-angle identity
The final step illustrates the importance of being able to recognize alterna-tive forms of identities.
Now try Exercise 27.
2 cos2 x 1 cos 2xcos 2x 2 cos2 x 1, so 1 cos 2x
2 cos2 x
cos x
sin x2 sin x cos x
cot x sin 2x cos x
sin x sin 2x
cot x sin 2x 1 cos 2x
tan 2 sin 2cos 2
2425
7
25
247
cos 2.sin 2tan 2
tan 2 2 tan
1 tan2
2431 169
83
79
247
tan sin cos
45
35
43.
tan 2
cos 2 cos2 sin2 9
25 1625
725
cos 2,
sin 45 ; cos 35 2 45 35 2425 .
sin 2 2 sin cos
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7.4 Double-Angle Identities and Half-Angle Identities 635
EXAMPLE 3 Simplifying Expressions Using Double-Angle Identities
Simplify each expression.
(a) (b)Solution
(a) This expression suggests one of the double-angle identities for cosine:Substituting 7x for A gives
(b) If this expression were 2 sin 15 cos 15, we could apply the identity forsin 2A directly since We can still apply the identitywith by writing the multiplicative identity element 1 as
Multiply by 1 in the form
Associative property (Section R.1)
with
(Section 5.3)
Now try Exercises 13 and 15.
Identities involving larger multiples of the variable can be derived by re-peated use of the double-angle identities and other identities.
EXAMPLE 4 Deriving a Multiple-Angle Identity
Write sin 3x in terms of sin x.
Solution
Sine sum identity(Section 7.3)
Double-angle identities
Multiply.
Distributive property(Section R.1)
Combine terms.
Now try Exercise 21.
3 sin x 4 sin3 x
2 sin x 2 sin3 x sin x sin3 x sin3 xcos2 x 1 sin2 x 2 sin x1 sin2 x 1 sin2 x sin x sin3 x
2 sin x cos2 x cos2 x sin x sin3 x 2 sin x cos x cos x cos2 x sin2 x sin x
sin 2x cos x cos 2x sin x sin 3x sin2x x
sin 30 12 12
12
14
12
sin 30
A 152 sin A cos A sin 2A, 12
sin2 15
12
2 sin 15 cos 15
12 2. sin 15 cos 15
12
2 sin 15 cos 15
12 2.A 15
sin 2A 2 sin A cos A.
cos2 7x sin2 7x cos 27x cos 14x.
cos 2A cos2 A sin2 A.
sin 15 cos 15cos2 7x sin2 7x
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636 CHAPTER 7 Trigonometric Identities and Equations
Looking Ahead to CalculusThe product-to-sum identities are usedin calculus to find integrals of func-tions that are products of trigonometricfunctions. One classic calculus text in-cludes the following example:
Evaluate
The first solution line reads:We may write
cos 5x cos 3x 12
cos 8x cos 2x.
cos 5x cos 3xdx.
2000
500
0 .05
For x = t,
W(t) = (163 sin 120t)2
15
Figure 6
The next example applies a multiple-angle identity to answer a questionabout electric current.
EXAMPLE 5 Determining Wattage Consumption
If a toaster is plugged into a common household outlet, the wattage consumed isnot constant. Instead, it varies at a high frequency according to the model
where V is the voltage and R is a constant that measures the resistance of the toaster in ohms. (Source: Bell, D., Fundamentals of Electric Circuits,Fourth Edition, Prentice-Hall, 1998.) Graph the wattage W consumed by a typi-cal toaster with and in the window by
How many oscillations are there?
Solution Substituting the given values into the wattage equation gives
To determine the range of W, we note that sin 120 t has maximum value 1, sothe expression for W has maximum value The minimum value is 0.The graph in Figure 6 shows that there are six oscillations.
Now try Exercise 81.
Product-to-Sum and Sum-to-Product Identities Because they make itpossible to rewrite a product as a sum, the identities for and
are used to derive a group of identities useful in calculus.Adding the identities for and gives
or
Similarly, subtracting from gives
Using the identities for and in the same way, we gettwo more identities. Those and the previous ones are now summarized.
Product-to-Sum Identities
(continued)
sin A sin B 12
[cos(A B) cos(A B)]
cos A cos B 12
[cos(A B) cos(A B)]
sinA BsinA B
sin A sin B 12
cosA B cosA B.
cosA BcosA B
cos A cos B 12
cosA B cosA B.
cosA B cosA B 2 cos A cos B cosA B cos A cos B sin A sin B
cosA B cos A cos B sin A sin BcosA BcosA B
cosA BcosA B
163215 1771.
W V 2
R
163 sin 120t2
15 .
500, 2000.0, .05V 163 sin 120tR 15
W V 2
R,
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7.4 Double-Angle Identities and Half-Angle Identities 637
EXAMPLE 6 Using a Product-to-Sum Identity
Write cos 2 sin as the sum or difference of two functions.
Solution Use the identity for cos A sin B, with and
Now try Exercise 41.
From these new identities we can derive another group of identities that areused to write sums of trigonometric functions as products.
Sum-to-Product Identities
EXAMPLE 7 Using a Sum-to-Product Identity
Write as a product of two functions.
Solution Use the identity for with and
(Section 7.1)
Now try Exercise 45.
sin sin 2 cos 3 sin 2 cos 3 sin
2 cos 62
sin22 sin 2 sin 4 2 cos2 42 sin2 42
4 B.2 Asin A sin B,
sin 2 sin 4
cos A cos B 2 sinA B2 sinA B2 cos A cos B 2 cosA B2 cosA B2 sin A sin B 2 cosA B2 sinA B2 sin A sin B 2 sinA B2 cosA B2
12
sin 3 12
sin
cos 2 sin 12
sin2 sin2
B.2 A
cos A sin B 12
[sin(A B) sin(A B)]
sin A cos B 12
[sin(A B) sin(A B)]
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638 CHAPTER 7 Trigonometric Identities and Equations
Half-Angle Identities From the alternative forms of the identity for cos 2A,we derive three additional identities for and These areknown as half-angle identities.
To derive the identity for start with the following double-angle iden-tity for cosine and solve for sin x.
Add subtract cos 2x.
Divide by 2; take square roots. (Section 1.4)
Let so substitute.
The sign in this identity indicates that the appropriate sign is chosen de-pending on the quadrant of For example, if is a quadrant III angle, wechoose the negative sign since the sine function is negative in quadrant III.
We derive the identity for using the double-angle identity
Add 1.
Rewrite; divide by 2.
Take square roots.
Replace x with
An identity for comes from the identities for and
We derive an alternative identity for using double-angle identities.
Double-angle identities
From this identity for we can also derive
tan A2
1 cos A
sin A.
tan A2 ,
tan A2
sin A
1 cos A
sin 2 A2
1 cos 2 A2
tan A2
sin A2cos
A2
2 sin A2 cos
A2
2 cos2 A2
tan A2
tan A2
sin A2cos
A2
1 cos A21 cos A2
1 cos A1 cos Acos
A2 .sin
A2tan
A2
A2 . cos
A2
1 cos A2 cos x 1 cos 2x2 cos2 x
1 cos 2x2
1 cos 2x 2 cos2 x
cos 2x 2 cos2 x 1.cos
A2
A2
A2 .
x A2 ;2x A, sin A2
1 cos A2 sin x 1 cos 2x2
2 sin2 x; 2 sin2 x 1 cos 2x cos 2x 1 2 sin2 x
sin A2 ,
tan A2 .cos A2 ,sin
A2 ,
Multiply by 2 in numer-ator and denominator.
cos A2
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7.4 Double-Angle Identities and Half-Angle Identities 639
TEACHING TIP Point out that the
first identity for follows
directly from the cosine and sinehalf-angle identities; however, the
other two identities for are
more useful.
tan A2
tan A2
TEACHING TIP Have students com-pare the value of cos 15 inExample 8 to the value in Example 1(a) of Section 7.3,where we used the identity for thecosine of the difference of twoangles. Although the expressionslook completely different, they areequal, as suggested by a calculatorapproximation for both,.96592583.
Half-Angle Identities
N O T E The last two identities for do not require a sign choice. Whenusing the other half-angle identities, select the plus or minus sign according tothe quadrant in which terminates. For example, if an angle then
which lies in quadrant II. In quadrant II, and are negative,while is positive.
EXAMPLE 8 Using a Half-Angle Identity to Find an Exact Value
Find the exact value of cos 15 using the half-angle identity for cosine.
Solution
Choose the positive square root.
Simplify the radicals. (Section R.7)
Now try Exercise 51.
EXAMPLE 9 Using a Half-Angle Identity to Find an Exact Value
Find the exact value of tan 22.5 using the identity
Solution Since replace A with 45.
Now multiply numerator and denominator by 2. Then rationalize the denominator.
Now try Exercise 53.
22 1
2 2 1
tan 22.5 22 2
2
2 2
2 22 2
22 2
2
tan 22.5 tan 452
sin 45
1 cos 45 22
1 22
22.5 12 45,
tan A2 sin A
1 cos A .
1 322
1 32 22 2
2 3
2
cos 15 cos 12
30 1 cos 302
sin A2
tan A2cos A2
A2 162,
A 324,A2
tan A2
tan A2
1 cos A1 cos A tan A2 sin A1 cos A tan A2 1 cos Asin Acos
A2
1 cos A2 sin A2 1 cos A2
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640 CHAPTER 7 Trigonometric Identities and Equations
Figure 7
0
32
2
s2
s
EXAMPLE 10 Finding Functions of Given Information about s
Given with find and
Solution Since
and Divide by 2. (Section 1.7)
terminates in quadrant II. See Figure 7. In quadrant II, the values of andare negative and the value of is positive. Now use the appropriate half-
angle identities and simplify the radicals.
Notice that it is not necessary to use a half-angle identity for once we findand However, using this identity would provide an excellent check.
Now try Exercise 59.
EXAMPLE 11 Simplifying Expressions Using the Half-Angle Identities
Simplify each expression.
(a) (b)
Solution
(a) This matches part of the identity for
Replace A with 12x to get
(b) Use the third identity for given earlier with to get
Now try Exercises 67 and 71.
1 cos 5sin 5 tan
52
.
A 5tan A2
1 cos 12x2 cos 12x2 cos 6x.
cos A2
1 cos A2cos
A2 .
1 cos 5sin 51 cos 12x2
cos s2 .sin
s2
tan s2
tan s
2
sin s2cos
s2
66
306
55
cos s
2 1 232 56 306
sin s
2 1 232 16 66
sin s2tan s2
cos s2
s2
3
4
s
2 ,
3
2
s 2
tan s2 .sin s2 ,cos
s2 ,
3
2 s 2,cos s
23 ,
s2
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7.4 Double-Angle Identities and Half-Angle Identities 641
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12. 13. 14.
15. 16.
17. 18.
19. 20.
21.22.
23.
24.25.
26. 4 tan x cos2 x 2 tan x
1 tan2 x sin 2x
cos4 x sin4 x cos 2xcos 4x 8 cos4 x 8 cos2 x 1
tan 3x 3 tan x tan3 x
1 3 tan2 x
4 sin3 x cos xsin 4x 4 sin x cos3 x cos 3x 4 cos3 x 3 cos x
116
sin 5914
cos 94.2
14
tan 6812
tan 102
24
22
22
32
33
32
sin 2 266
25
cos 2 1925;
sin 2 455
49
cos 2 3949
;
sin 2x 1517
cos 2x 8
17;
sin 2x 45cos 2x
35 ;
sin 2 120169
cos 2 119169
;
sin 2 421
25
cos 2 1725;
sin x 6
6cos x
306
;
sin x 102
12cos x
4212
;
sin 2
4cos
144
;
sin 5
5cos 25
5 ;
7.4 Exercises
Use identities to find values of the sine and cosine functions for each angle measure. SeeExample 1.
1. , given and terminates in quadrant I
2. , given and terminates in quadrant III
3. given and
4. given and
5. , given and
6. , given and
7. given and
8. given and
9. given and
10. , given and
Use an identity to write each expression as a single trigonometric function value or as asingle number. See Example 3.
11. 12. 13.
14. 15. 16.
17. 18. 19.
20.
Express each function as a trigonometric function of x. See Example 4.21. 22. 23. 24.
Graph each expression and use the graph to conjecture an identity. Then verify yourconjecture algebraically.
25. 26.
Verify that each equation is an identity. See Example 2.
27. 28. sec 2x sec2 x sec4 x
2 sec2 x sec4 xsin x cos x2 sin 2x 1
4 tan x cos2 x 2 tan x1 tan2 x
cos4 x sin4 x
cos 4xtan 3xsin 4xcos 3x
18
sin 29.5 cos 29.5
14
12
sin2 47.1tan 34
21 tan2 34tan 51
1 tan2 51
cos2
8
12
2 cos2 6712
11 2 sin2 2212
1 2 sin2 152 tan 151 tan2 15cos
2 15 sin2 15
sin 0cos 3
52
cos 0sin 5
72,
sin x 0tan x 53
2x,
cos x 0tan x 22x,
sin 0cos 1213
2
cos 0sin 252
2 x cos 2x
23
x,
2 x cos 2x
512
x,
cos 2 34
cos 2 35
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642 CHAPTER 7 Trigonometric Identities and Equations
41.42.
43.
44.
45.46.47.48.49.50.
51. 52.
53. 54.
55. 56.
59. 60. 61. 3
62.
63.
64.
65. 66. 557
50 151010
50 10510
50 206
10
134
104
2 32
2 3
2
2 32 3
2
2 3
22 2
2
2 cos 6x sin 3x2 cos 6x cos 2x2 cos 98.5 sin 3.52 sin 11.5 cos 36.52 cos 6.5x cos 1.5x2 sin 3x sin x
12
cos x 12
cos 9x
52
cos 5x 52
cos x
sin 225 sin 55sin 160 sin 44
9
29. 30.
31. 32.
33. 34.
35. 36.
37. 38.
39. 40.
Write each expression as a sum or difference of trigonometric functions. See Example 6.41. 42.43. 44.
Write each expression as a product of trigonometric functions. See Example 7.45. 46. 47.48. 49. 50.
Use a half-angle identity to find each exact value. See Examples 8 and 9.51. 52. 53.54. 55. 56.57. Explain how you could use an identity of this section to find the exact value of
(Hint: .)58. The identity can be used to find and
the identity can be used to find . Show thatthese answers are the same, without using a calculator. (Hint: If and and then )
Find each of the following. See Example 10.
59. given with
60. given with
61. given with
62. given with
63. given , with
64. given with
65. given with
66. given with 90 180tan 52
,cot
2,
180 270tan 7
3,tan
2,
2 x cot x 3,cos
x
2,
0 x
2tan x 2sin
x
2,
180 270sin 15 ,cos
2,
90 180sin 35 ,tan
2,
2 x cos x
58
,sin x
2,
0 x
2cos x
14
,cos x
2,
a b.a2 b2,b 0a 0
tan 22.5 2 1tan A2 sin A
1 cos A
tan 22.5 3 22,tan A2 1 cos A1 cos A7.5 121230sin 7.5.
sin 165cos 165tan 195cos 195sin 195sin 67.5
sin 9x sin 3xcos 4x cos 8xsin 102 sin 95sin 25 sin48cos 5x cos 8xcos 4x cos 2x
sin 4x sin 5x5 cos 3x cos 2x2 cos 85 sin 1402 sin 58 cos 102
sin 2x2 sin x
cos2 x
2 sin2
x
2sin2
x
2
tan x sin x2 tan x
cot2 x
2
1 cos x2
sin2 xsec2
x
2
21 cos x
cot x tan xcot x tan x
cos 2xtan x cot x 2 csc 2x
cos 2x 1 tan2 x1 tan2 x
sin 2x cos 2x sin 2x 4 sin3 x cos x
sin 4x 4 sin x cos x 8 sin3 x cos x2 cos 2xsin 2x
cot x tan x
1 cos 2xsin 2x
cot xsin 4x 4 sin x cos x cos 2x
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Use an identity to write each expression with a single trigonometric function. SeeExample 11.
67. 68. 69.
70. 71. 72.
73. Use the identity to derive the equivalent identity by multiplying both the numerator and denominator by
74. Consider the expression (a) Why cant we use the identity for to express it as a function of x alone?(b) Use the identity to rewrite the expression in terms of sine and
cosine.(c) Use the result of part (b) to show that
(Modeling) Mach Number An airplane flying fasterthan sound sends out sound waves that form a cone, asshown in the figure. The cone intersects the ground toform a hyperbola. As this hyperbola passes over a par-ticular point on the ground, a sonic boom is heard at thatpoint. If is the angle at the vertex of the cone, then
where m is the Mach number for the speed of the plane. (We assume .) The Machnumber is the ratio of the speed of the plane and the speed of sound. Thus, a speed ofMach 1.4 means that the plane is flying at 1.4 times the speed of sound. In Ex-ercises 7578, one of the values or m is given. Find the other value.75. 76. 77. 78.
(Modeling) Solve each problem. See Example 5.79. Railroad Curves In the United States, circular rail-
road curves are designated by the degree of curvature,the central angle subtended by a chord of 100 ft. Seethe figure. (Source: Hay, W. W., Railroad Engineering,John Wiley & Sons, 1982.)(a) Use the figure to write an expression for (b) Use the result of part (a) and the third half-angle
identity for tangent to write an expression for (c) If what is the measure of angle to the nearest degree?
80. Distance Traveled by a Stone The distance Dof an object thrown (or propelled) from height h(feet) at angle with initial velocity v is modeledby the formula
See the figure. (Source: Kreighbaum, E. and K. Barthels, Biomechanics, Allyn &Bacon, 1996.) Also see the Chapter 5 Quantitative Reasoning.(a) Find D when that is, when the object is propelled from the ground.(b) Suppose a car driving over loose gravel kicks up a small stone at a velocity of
36 ft per sec (about 25 mph) and an angle . How far will the stone travel? 30
h 0;
D v2 sin cos v cos v sin 2 64h
32.
b 12,tan 4 .
cos 2 .
60 30m 54
m 32
m 1
sin
2
1m
,
tan2 x cot x.
tan sin cos
tanA Btan2 x.
1 cos A.tan A2
1 cos Asin Atan
A2
sin A1 cos A
sin 158.21 cos 158.2
1 cos 59.74sin 59.741 cos 1651 cos 165
1 cos 1471 cos 1471 cos 7621 cos 402
7.4 Double-Angle Identities and Half-Angle Identities 643
67. 68.69. 70.71. 72.
74. (a) is undefined, so it cannot be used.
(b)
75. 84 76. 106 77. 3.9
78. 2 79. (a)
(b) (c) 54
80. (a)(b) approximately 35 ft
D v2 sin 2
32
tan
4
b50
cos
2
R bR
tan 2 x sin 2 xcos 2 x
tan
2
tan 79.1tan 29.87cot 82.5tan 73.5
cos 38sin 20
50 50b
R2
2
h
D
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81. Wattage Consumption Refer to Example 5. Use an identity to determine values ofa, c, and so that Check your answer by graphing both expres-sions for W on the same coordinate axes.
82. Amperage, Wattage, and Voltage Amperage is a measure of the amount of elec-tricity that is moving through a circuit, whereas voltage is a measure of the forcepushing the electricity. The wattage W consumed by an electrical device can bedetermined by calculating the product of the amperage I and voltage V. (Source:Wilcox, G. and C. Hesselberth, Electricity for Engineering Technology, Allyn &Bacon, 1970.)(a) A household circuit has voltage when an incandescent
lightbulb is turned on with amperage Graph the wattageconsumed by the lightbulb in the window by
(b) Determine the maximum and minimum wattages used by the lightbulb.(c) Use identities to determine values for a, c, and so that (d) Check your answer by graphing both expressions for W on the same coordinate axes.(e) Use the graph to estimate the average wattage used by the light. For how many
watts do you think this incandescent lightbulb is rated?
W a cost c.
50, 300.0, .05W VII 1.23 sin120t.
V 163 sin120t
W a cost c.
644 CHAPTER 7 Trigonometric Identities and Equations
81. , ,
82. (a)
(b) maximum: 200.49 watts;minimum: 0 watts(c) , ,
(e) 100.245 wattsc 100.245 240a 100.245
300
50
0 .05
For x = t, W = VI =(163 sin 120t)(1.23 sin 120t)
240
c 885.6a 885.6
These summary exercises provide practice with the various types of trigonometric iden-tities presented in this chapter. Verify that each equation is an identity.1. 2.
3. 4.
5. 6.
7. 8.
9. 10.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
21. 22.tan cot tan cot
1 2 cos2 cot s tan scos s sin s
cos s sin s
sin s cos s
tan x2 4 sec x tan xtan 4 2 tan 22 sec2 2sin s
1 cos s
1 cos ssin s
2 csc stan2 t 1tan t csc2 t
tan t
cos 2x 2 sec2 x
sec2 xcos x
1 tan2 x21 tan2 x2
csc4 x cot4 x 1 cos2 x1 cos2 x
sin tan 1 cos
tan
1 tan2
2
2 cos 1 cos
sinx ycosx y
cot x cot y
1 cot x cot y
1sec t 1
1
sec t 1 2 cot t csc tcot tan
2 cos2 1sin cos
21 cos x
tan2 x
2 1sin 2
2 tan 1 tan2
1 sin tcos t
1
sec t tan tsin t
1 cos t
1 cos tsin t
sec x sec xtan x
2 csc x cot x
csc cos2 sin csc tan cot sec csc
Summary Exercises on Verifying Trigonometric Identities
LIALMC07_0321227638.QXP 2/26/04 10:47 AM Page 644
23. 24.
25. 26.
27. 28. 12
cot x
2
12
tan x
2 cot x
2sin x sin3 xcos x
sin 2x
csc t 1csc t 1
sec t tan t2cos4 x sin4 x
cos2 x 1 tan2 x
2 cos2 x
2 tan x tan x sin x
tanx y tan y1 tanx y tan y
tan x
7.5 Inverse Circular Functions 645
Inverse Functions We first discussed inverse functions in Section 4.1. Wegive a quick review here for a pair of inverse functions f and
1. If a function f is one-to-one, then f has an inverse function 2. In a one-to-one function, each x-value corresponds to only one y-value and
each y-value corresponds to only one x-value.3. The domain of f is the range of and the range of f is the domain of 4. The graphs of f and are reflections of each other about the line 5. To find from follow these steps.
Step 1 Replace with y and interchange x and y.Step 2 Solve for y.Step 3 Replace y with
In the remainder of this section, we use these facts to develop the inversecircular (trigonometric) functions.Inverse Sine Function From Figure 8 and the horizontal line test, we seethat does not define a one-to-one function. If we restrict the domain tothe interval which is the part of the graph in Figure 8 shown in color,this restricted function is one-to-one and has an inverse function. The range of
is so the domain of the inverse function will be and its range will be
Figure 8
y
1
(0, 0)1
2 y = sin x
Restricted domain
x
2 2
32
32
2
2
[ , ]2 2
( , 1)2
( , 1)2
2 , 2 .1, 1,1, 1,y sin x
2 , 2 ,y sin x
f 1x.
fxfx,f 1x
y x.f 1f 1.f 1,
f 1.f 1.
7.5 Inverse Circular FunctionsInverse Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Remaining Inverse Circular Functions Inverse Function Values
Looking Ahead to CalculusThe inverse circular functions are usedin calculus to solve certain types of re-lated-rates problems and to integratecertain rational functions.
TEACHING TIP Mention that the inter-
val contains enough of
the graph of the sine function toinclude all possible values of y. Whileother intervals could also be used,this interval is an accepted conventionthat is adopted by scientific calcula-tors and graphing calculators.
2 , 2
LIALMC07_0321227638.QXP 2/26/04 10:47 AM Page 645
646 CHAPTER 7 Trigonometric Identities and Equations
Algebraic Solution
(a) The graph of the function defined by (Figure 9) includes the point Thus,
Alternatively, we can think of as y is the number in whose sine is Then we can write the given equation as Since and is in the range of the arcsine function,
(b) Writing the equation in the formshows that This can be veri-
fied by noticing that the point is on thegraph of
(c) Because is not in the domain of the inversesine function, does not exist.
Graphing Calculator Solution
To find these values with a graphing calculator, wegraph and locate the points with X-values
and Figure 10(a) shows that when Similarly, Figure 10(b) shows
that when
(a) (b)
Figure 10
Since does not exist, a calculator will givean error message for this input.
sin12
2
2
1 1
2
2
1 1
Y 2 1.570796.X 1,Y 6 .52359878.
X 12 ,1.12
Y1 sin1 X
sin122
y sin1 x.1, 2
y 2 .sin y 1y sin11
y 6 .
6sin
6
12
sin y 12 .
12 .2 , 2
y arcsin 12
arcsin 12
6.
12 , 6 .y arcsin x
Reflecting the graph of on the restricted domain across the line gives the graph of the inverse function, shown in Figure 9. Some key points are labeled on the graph. The equation of the inverse of isfound by interchanging x and y to get This equation is solved for y bywriting (read inverse sine of x). As Figure 9 shows, the domain of
is while the restricted domain of is therange of An alternative notation for is
Inverse Sine Function
or means that for
We can think of or as y is the number in the inter-val whose sine is x. Just as we evaluated by writing it in ex-ponential form as (Section 4.3), we can write as to evaluate it. We must pay close attention to the domain and range intervals.
EXAMPLE 1 Finding Inverse Sine Values
Find y in each equation.
(a) (b) (c) y sin12y sin11y arcsin 12
sin y xy sin1 x2y 4y log2 42 , 2
y arcsin xy sin1 x
2 y
2 .x sin y,y arcsin xy sin1 x
arcsin x.sin1 xy sin1 x.2 , 2 ,y sin x,1, 1,y sin1 x
y sin1 xx sin y.
y sin xy x
y sin x
Now try Exercises 13 and 23.
C A U T I O N In Example 1(b), it is tempting to give the value of as since Notice, however, that is not in the range of the inverse
sine function. Be certain that the number given for an inverse function value isin the range of the particular inverse function being considered.
3
2sin
3
2 1.
3
2 ,
sin11
1
0
y = sin1 x or y = arcsin x
1
(0, 0)
y
x
(1, )2
(1, )
2
2
3( , )32
4( , )22
4( , )22
3( , )32
612( , )
612( , )
2
Figure 9
LIALMC07_0321227638.QXP 2/26/04 10:47 AM Page 646
Our observations about the inverse sine function from Figure 9 lead to thefollowing generalizations.
INVERSE SINE FUNCTIONy sin1x or y arcsin x
Domain: Range:
Figure 11
The inverse sine function is increasing and continuous on its domain
Its x-intercept is 0, and its y-intercept is 0. Its graph is symmetric with respect to the origin; it is an odd function.
Inverse Cosine Function The function (or ) is de-fined by restricting the domain of the function to the interval asin Figure 12, and then interchanging the roles of x and y. The graph of
is shown in Figure 13. Again, some key points are shown on thegraph.
Figure 12 Figure 13
Inverse Cosine Function
or means that for 0 y .x cos y,y arccos xy cos1 x
y
x
10
y = cos1 x or y = arccos x
(1, 0)1
(1, )
6( , )32
4( , )22
312( , )
23
( , )22 34( , )32 56
12( , )(0, )2
y
1
0
y = cos xRestricted domain
[0, ]
(0, 1)
(, 1)
1
x( , 0)2
2