Section 5.2 Verifying Trigonometric Identities

[Pages:6]251 PART I: Solutions to Odd-Numbered Exercises and Practice Tests

127. sinA =-a==> a = c. sin A = 20 sin 28? ~ 9.39

c

B = 90? -A? = 62?

cosA=-b==~ b=c.cosA~ 17.66

c

129. a = ~/c2 - b2 = -~/12.542 - 6.22 ~ 10.90 sin B .bc..1. 2=6.=.52~4 B ~ 29.63? A = 90? - 29.63? = 60.37?

Section 5.2 Verifying Trigonometric Identities

[] You should know the difference between an expression, a conditional equation, and an identity. [] You should be able to solve trigonometric identities, using the following techniques.

(a) Work with one side at a time. Do not "cross" the equal sign. (b) Use algebraic techniques such as combining fractions, factoring expressions, rationalizing denominators,

and squaring binomials. (c) Use the fundamental identities. (d) Convert all the terms into sines and cosines.

Solutions to Odd-Numbered Exercises

1. sin t csc t = sin

= 1

csc2 x 1 sin x cot X sinE x cos x

CSCX " secx

1 sin x ? cos x

5. COSEft- sinEft= (1 -sinEft)- sin2fl = 1 -- 2 sinEfl

7. tanEo+6=(secE0- 1)+6

= sec2 19 + 5

sin x 9. cos x + sin x tan x = cos x + sin x ? ~

cos x

cos2 x + sin2 x

COS X

1

COS X

= sec X

11. x

Yl

?Y2

0.2 4.835 4.835

0.4 0.6 0.8 1.0 1.2 1.4 2.1785 1.2064 0.6767 0.3469 0.1409 0.0293 2.1785 1.2064 0.6767 0.3469 0.1409 0.0293

1

cos x

sec x tan x = COS Xs*in~x

COS2 x

sin x 1 - sin2 x

sin x

1 sin x

sin x

= csc x - sin x

252 PART 1: Solutions to Odd-Numbered Exercises and Practice Tests

13. x

Yl Y2

0.2 4.835 4.835

0.4 0.6 0.8 1.0 1.2 1.4 2.1785 1.2064 0.6767 0.3469 0.1409 0.0293 2.1785 1.2064 0.6767 0.3469 0.1409 0.0293

o o

cscx

-

1 sinx s=in x~-

sin x

1 - sin2 x

sin x

cos2 x

sin x

=

COS

X

?

COS x

sin x

-" COSX " cotx

15. i X

Yl Y2

0.2 0.4 5,0335 2.5679 5.0335 2.5679

0.6 1.7710 1.7710

0.8 1.3940 1.3940

1.0 1.1884 1.1884

1.2 1.0729 1.0729

1.4 1.0148 1.0148

sin

x

+

cos

x

cot

x

=

sin

x

+

cos

'xCs.OmS

X

x

_,_ sin2 x + cos2 x sin x

1 sin x

= csc x

1.5

17. x 0.2 0.4 0.6 0.8 Yl 5.1359 2.7880 2.1458 Y2 5.1359 2.7880 2.i458

1.0 1.2 1.4 2.1995 2.9609 5.9704 2.1995 2.9609 5.9704

~1 +~1 =c?tx+tanx tanx cotx tanx.cotx

= cot x + tan x

oo

19. The error is in line 1: cot(- x) #'cot x.

21. Missing step: (sec2x - 1)2 = (tan2 X)2 -" tarl4 x

23. sin~/2 x cos x - sin~/2 x cos x = sin1/2 x cos x(1 - sin2 x) = sint/2 x cos x ? cos2 x = coss x.,/~ffx

-- X SeCX = COtX" secx

1

27. scees(c-(x-.x__)._..~) = c1os(-x) =csoins((--xx))

- ~sin=x- tan x

COS X

253 PART 1: Solutions to Odd-Numbered Exercises and Practice Tests

29.

cos(-0)

1 + sin(-0)

cos 0 1 + sin 0 1-sin0 1 +sin0

cos ~(1 + sin 0) 1 - sin2 0_

cos ~(1 + sin O) cos2 ~

1 + sin O

~os 0

1 sin 0

cos 0 cos O

= sec 0 + tan 0

31.

sin x cos y + cos x sin y cos x cos y - sin x sin y

sin x cos cos x Cos y

y +

cos x cos x

sin cos

y y

cos x cos y sin x sin y cos x cos y cos x cos y

tan x + tan y 1 - tan x tan y

1

1 tan y + cot x

33.

tan x + cot y -tan x cot y

cot x tan y cot x ? tan y

1 1

1

= tan y + cot x

cot

o

x tan

y

cot

x

?

tan

y

35. ~/~+sinO_ ~/~+sinO. l+sinO -sinO -sinO 1 +sinO + sin 0)2 =~/(i -- sin2 0

Note: Check your answer with a graphing utility. What happens if you leave off the absolute value?

cos2 O 1 + sin 0

37. cos2x+cos -x =cos2x+sin2x= 1

?

-x =secx.cosx= 1

41. 2 sec2 x -- 2 secx x sin2 x - sin2 x -- COS2 x = 2 sec2 x(1 - sin2x) - (sin2 x + cos2 x) = 2 sec2 x(cos2 x) -- 1 1 =2"~"ccoos2s2xx- 1 =2-1 =1

43. 2 + cos2x - 3 cos'ix = (1 - cos2 x)(2 + 3 cos2 x) = sin2 x(2 + 3 cos2 x)

254 PART I: Solutions to Odd-Numbered Exercises and Practice Tests

45. csc4 x - 2 csc2 x + 1 = (csc2 x - 1)2 = (cot2 x)2 = cot4 x

47. see4 0 - tan4 0 = (see2 0 + tall20)(sec2 0 -- tan2 O) -- (1 + tanz 0 + tan2 0)(1) = 1 + 2tan20

49.

sin/3 1 -cos/3

1 + cos/3 1 +cos/3

--

sin f!(1 + cos f!) 1 - cos2/3

sin/3(1 + cos/3) 1 + cos/3

sin2/3

--

sin/3

51. ttaann3aa--11 (-tan a - 1)(tatna2naa+- t1an a + ~1) =tan2a+tana+ 1

53. It appears that Yl = 1. Analytically, 1 +1 tanx-+ 1 +cotx+ 1

cotx + 1 tan x + 1 (cot x + 1)(tanx + 1)

.--.

tanx + cotx + 2

cotxtanx + cotx + tanx + 1

tan x -t- cot x ? 2 tanx+cotx+2

=1.

55. It appears that Yl -" Sin X. Analytically,

1

COS2 X --

1 -- COS2 X

--

sin2x - sinx.

sin x sin x sin x sinx

2

57. Inlcot 01 - _ Icos 0

- tnlsin 01 - lnlcos 0l - Inlsin 01

-2

59. -ln(1 + cos 0) = In(1 + cos 0)-1

= lnl +cosO 1 -cos

=ln

1 1

-

cos 0 cos2 0

=In

1

- cos sin2 0

0

= In(1 - cos 0) - In sin2 0

= In(1 - cos o) - 2 Inlsin ol

61. sin2 25? + sin2 65? = sin2 25? + cos2 25? = 1

255 PART 1: Solutions to Odd-Numbered Exercises and Practice Tests

63. cos2 20? + cos2 52? + cos~ 38? + cos2 70? = c?s2 20? + c?s2 522 + sin2(90? - 38?) + sin2(90? - 70?) = cos2 20? + cos~ 522 + sin252? + sinz 20? = (cos2 20? + sin2 20?) + (cos~ 52? + sin~ 52?) =1+1 =2

65. tanS x = tanax " tan2 x = tan3 x(sec2 x - 1) = tan3 x sec2 x -- tan3 x

67. (sinz x - sin~ x)cos x = sin~ x(1 - sinz x)cos x

-- sin2X ? COS2 X ? COS X = COS3 X sin2 x

69. /zW cos 0 = W sin 0

W sin 0 sin 0 /z W cos 0 cos 0

tan 0, W 4:0

71. cos x - csc x. cot x = cos x

1 cos x

sin x sin x

cos

sina

- cos x(1 - csc2 x)

= cos x(-cot~ x)

73. True. f(x) = cos x and g(x) = sec x are even

75. False. For example, sin(l,z) 4: sinz (1)

79? ~/sin~ x + cos2x 4: sin x + cos x The left side is 1 for any x, but the right side is not necessarily 1. For example, the equation is not tree for x = 7r/4.

+ 1)~! = sin[51-(12n~r + ~r)l $1. sinI(12n 6

= sin(2n,rr + -~) = sin"6t2-r 1 Thus, sin[(126n ? 1?)qr=]~1for all integers

83. (x- i)(x + i)(x- 4i)(x + 40 = (x~ + 1)(x2 + 16) =x*+ 17xz+ 16

256 PART 1: Solutions to Odd-Numbered Exercises and Practice Tests

87. f(x) = -2x-3

y

89, f(x) = 5-x - 2

y

2 4 6

91. s = rO

0

-

s r

12-16

~ 2.3636 radians

93. Quadrant III

95. Quadrant III

Section 5.3 Solving Trigonometric Equations

[] You should be able to identify and solve trigonometric equations. [] A trigonometric equation is a conditional equation. It is true for a specific set of values. [] To solve trigonometric equations, use algebraic techniques such as collecting like terms, taking square roots,

factoring, squaring, converting to quadratic form, using formulas, and using inverse functions. Study the examples in this section. I Use your graphing utility to calculate solutions and verify results.

Solutions to Odd-Numbered Exercises 1. 2cosx- 1 =0

(a) 2cos~- 1 =2 - 1 =0 (b) 2cos'~'-- 1 = 2 - 1 =0

3. 3tan22x- 1 =0

(a) 3 tan\-~-/j - 1 = 3~an2-~- 1

=3

-1

=0

(b) [3(ltOa'nnk']]212~_]~J -1 =3tan~ - 1

=0

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