Verifying Trigonometric Identities

Transcription

1 25 PART I: Solutions to Odd-Numbered Exercises and Practice Tests a 27. sina =- ==> a = c. sin A = 20 sin 28 ~ 9.39 c B = 90 -A = 62 b cosa=- ==~ b=c.cosa~ 7.66 c 29. a = ~/c 2 - b 2 = -~/ ~ 0.90 b 6.2 sin B... ==~ B ~ c 2.54 A = = 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. sin t csc t = sin = csc 2 x cot X sin E x cos x cos x CSCX " secx 5. COS Eft- sin Eft= ( -sin Eft)- sin 2fl = 2 sinefl 7. tan Eo+6=(secE0- )+6 = sec cos x + tan x = cos x + ~ cos 2 x + sin 2 x COS X = sec X COS X cos x. x Yl Y sec x tan x cos x = COS X * ~ COS 2 x - sin 2 x = csc x -

2 252 PART : Solutions to Odd-Numbered Exercises and Practice Tests 3. x Yl Y cscx - sinx = ~- - sin 2 x cos 2 x COS x = COS X -" COSX " cotx o o 5. i X Yl 5, Y COS X + cos x cot x = + cos x ṡm x _,_ sin 2 x + cos 2 x = csc x.5 7. x ~ +~ =c tx+tanx tanx cotx tanx.cotx Yl = cot x + tan x Y i o o 9. The error is in line : cot(- x) # cot x. 2. Missing step: (sec2x - ) 2 = (tan 2 X) 2 -" tarl 4 x 23. sin ~/2 x cos x - sin ~/2 x cos x = sin /2 x cos x( - sin 2 x) = sin t/2 x cos x cos 2 x = cos s x.,/~ffx X SeCX = COtX" secx 27. see(-x.._..~) = cos(-x) = sin(-x) csc(-x) cos(-x) - ~ = - tan x COS X

3 29. cos(-0) + sin(-0) 253 PART : Solutions to Odd-Numbered Exercises and Practice Tests cos 0 + sin 0 -sin0 +sin0 cos ~( + sin 0) - sin 2 0_ cos ~( + sin O) cos 2 ~ + sin O ~os 0 sin 0 cos 0 cos O = sec 0 + tan 0 3. cos y + cos x sin y cos x cos y - sin y cos y + cos x sin y cos x Cos y cos x cos y cos x cos y sin y cos x cos y cos x cos y tan x + tan y - tan x tan y 33. tan x + cot y tan x cot y tan y + cot x cot x tan y cot x tan y = tan y + cot x o cot x tan y cot x tan y 35. ~/~+sino_ ~/~+sino. l+sino -sino -sino +sino =~/(i + sin 0)2 sin 2 0 Note: Check your answer with a graphing utility. What happens if you leave off the absolute value? + sin 0 cos 2 O 37. cos 2x+cos -x =cos 2x+sin2x= -x =secx.cosx= 4. 2 sec 2 x 2 sec x x sin 2 x - sin 2 x COS 2 x = 2 sec 2 x( - sin2x) - (sin 2 x + cos 2 x) = 2 sec 2 x(cos 2 x) =2"~" cos2x- cos 2 x =2- = cos2x - 3 cos ix = ( - cos 2 x)(2 + 3 cos 2 x) = sin 2 x(2 + 3 cos 2 x)

4 254 PART I: Solutions to Odd-Numbered Exercises and Practice Tests 45. csc 4 x - 2 csc 2 x + = (csc 2 x - ) see tan 4 0 = (see tall 20)(sec2 0 tan 2 O) = (cot 2 x) 2 = cot 4 x ( + tan z 0 + tan 2 0)() = + 2tan sin/3 + cos/3 -cos/3 +cos/3 sin f!( + cos f!) - cos 2/3 sin/3( + cos/3) sin 2/3 + cos/3 sin/3 5. tan3 a - (tan - a - )(tan 2 a + tan a + ~) =tan2a+tana+ tana- tana- 53. It appears that Yl =. Analytically, tanx+ +cotx+ + - cotx + tan x + (cot x + )(tanx + ).. =. tanx + cotx + 2 cotxtanx + cotx + tanx + tan x -t- cot x 2 tanx+cotx It appears that Yl -" Sin X. Analytically, COS 2 X COS 2 X 2 sin2x sinx - sinx. _ Icos Inlcot tnlsin 0 - lnlcos 0l - Inlsin ln( + cos 0) = In( + cos 0) - 6. sin sin 2 65 = sin cos 2 25 = = lnl +coso -cos - cos 0 =ln - cos cos 0 =In sin 2 0 = In( - cos 0) - In sin 2 0 = In( - cos o) - 2 Inlsin ol

5 255 PART : Solutions to Odd-Numbered Exercises and Practice Tests 63. cos cos cos ~ 38 + cos2 70 = c s c s sin2( ) + sin2( ) = cos cos ~ sin252 + sin z 20 = (cos sin 2 20 ) + (cos ~ 52 + sin ~ 52 ) =+ =2 65. tans x = tanax " tan 2 x = tan 3 x(sec 2 x - ) = tan 3 x sec 2 x tan 3 x 67. (sin z x - sin ~ x)cos x = sin ~ x( - sin z x)cos x sin 2X COS 2 X COS X = COS 3 X sin 2 x 69. /zw cos 0 = W sin 0 W sin 0 sin 0 /z W cos 0 cos 0 tan 0, W 4:0 7. cos x - csc x. cot x = cos x cos cos x sin a - cos x( - csc 2 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: sin z () 79 ~/sin ~ x + cos2x 4: + cos x The left side is for any x, but the right side is not necessarily. For example, the equation is not tree for x = 7r/4. + )~! = sin[5-(2n~r + ~r)l $. sini(2n 6 = sin(2n,rr + -~) "tr = sin Thus, sin[ (2n )qr] = ~ 6 for all integers 83. (x- i)(x + i)(x- 4i)(x + 40 = (x ~ + )(x 2 + 6) =x*+ 7x z+ 6

6 256 PART : Solutions to Odd-Numbered Exercises and Practice Tests 87. f(x) = -2 x-3 y 89, f(x) = 5 -x - 2 y s = ro s ~ radians r 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. 2cosx- =0 (a) 2cos~- =2 - =0 (b) 2cos ~ = 2 - =0 3. 3tan 22x- =0 (a) 3 tan\-~-/j - = 3~an2-~- =3 - =0 [ (lo n ]] 2 ~_~ (b) 3 tan k 2]J - =3tan ~ - =0