Seclion 5.3 Solving Trigonometric Equations

[Pages:8]256 PART I: Solutions to Odd-Numbered Exercises and Practice Tests 89. f(x) = 5-x - 2

y

4 -2

91. s = rO

0

....sr

26 11

~ 2.3636 radians

93. Qua&ant III

95. Qua&ant III

Seclion 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. [] Use your graphing utility to calculate solutions and verify results.

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

(b) 2 cos--2-- 1 = 2 - 1 ---0

3. 3tan22x- 1=0

(a) 3 tan \-~2] - 1 = 3 ~tn2-~- 1

=3

-1

=0

(b) 3Itan (10~r/q2\'~'-]1 - 1 = 3 tan2~-~- 1

__

=0

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

5. 2cos2x+ 3cosx+ 1=0 (a) 2cos2/'~4"-rr~'~-) +3co(s4"Trr)+ 1 = 2(-_~)2-(3_+~1) =0 (b) 2cos2"rr+ 3cos~+ 1 = 2(-1)2- 3 + 1 = 0

7. y = sin-~-+ 1

From the graph in the textbook we see that the curve has x-intercepts at x = -1 and at

J

9. y = tan t--~--~ - 3 From the graph in the textbook we see that the curve has x-intercepts at x = +2.

11. 2cosx + 1 = 0

2 cos x = -1

1 COS X -- ----2

2?r 3 X "-- ~

4qr

orx = ~

3

13. ~secx-2=0 .!g sec x = 2 sec x - 2 ,/~

cos x -- 2 x =6=orx =ll6~~r

15. 3csc2x-4=O

CSC2

X

4 ----3

sinx= ? 2 "rr 2~" 4"rr 5~r

X-- 3'3'3'3

17. 2sin22x= 1

1 ./2

sin2x=+-~=? 2

2x =3"Tn" ,z =5T"tr,2x =7T"rr,

9'rr 11o 13rr 15~"

2x=~---, 2x=---~---,2x- 4 , 2x= 4

,rr 3,n" 5qr 7~r 9"rr ll,tr.13?r 15"n"

Thus, X -- 8'8'8'8'8'8'8'8

(8 solutions).

19. 4cos2x- 3 =0

COS2

X

:

3 --4

COS X ---- i 2

,rr 5"rr 7'rr ll~r 6'6'6' 6

21.

sin2 x = 3 cos2 x

sin2x - 3(1 - sin2x) = 0

4 sin2 x = 3

sinx= ? 2 "rr 2"rr 4~r 5"rr

X-- 3'3'3'3

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

23. (3 tan2 x - 1)(tan2 x - 3) =0

3tan2x- 1=0

or

1 tanx = +~r~

~r 7~ X "-" ~ ~6'6

5~r ll,rr

orx= 6' 6

tan2 x - 3 = 0

tan x = + V'~

"n" 4,rr X ~- ~ ~3'3

2"n" 5,rr or X -- 3'3

25.

COS3X "- COSX

COS3X -- COSX -- 0

cos x(cos~ x - 1) - 0

cosx=O

or cos2x- 1 =0

qr 3~r x-2,2

COSX = +__1

x=O,

27. 3 tanax - tan x = 0 tan x(3 tanz x - 1) = 0

tanx=O or 3tan2x- 1=0

x = O, ,rr

tan x = +~/3

3 ?r 5~" 7"? 11~" x-6, 6, 6,6

29. sec2x - secx - 2 = 0

(sec x - 2)(sec x + 1) = 0

sec x - 2 = 0

or sec x + 1 = 0

sec x = 2

SeC X -- -- 1

"n" 5,0"

3'3

33.

cscx+cotx= 1

1 cos x

sin x sin x

1 +cosx=sinx

(1 + cos x)2 = sin2 x

1 +2cosx+cosZx= 1-cos2x

2cos2x + 2cosx = 0

2 cos x(cos x + 1) = 0

cosx=O qr

or cosx=-I

2' 2

(3?r/2 is extraneous.) (,r is extraneous.) x = ?r/2 is the only solution.

31. 2sinx+cscx=O 2 sin x +1~ = 0 sin x 2sin2x+ 1=0 Since 2 sin2 x + 1 > O, there are no solutions.

35. cos - 2 x ~- + 2n~r 2 4 x =~ +4nqr

37.

1

1

+--cc0ooss

x

x

1 + cos x = 0

cosx - -1

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

39.

2 secax + tanax - 3 = 0

2(tan2x + 1) + tan2x - 3 =0

3tan~x- 1-0

tan x = ~ 3

?r 5?r 7~r 117r X-" 6'6'6' 6

41.

sec2x q- tan x = 3

? (1 +tan~x)+tanx=3

tan2x + tanx -- 2 = 0

(tan x + 2)(tan x - 1) = 0

tanx=--2 or tanx= 1

43. y=9cosx- 1 x ~ 4.8237, 1.4595

6.28

-IO

45. y=4sin2x-2cosx- 1 x ~ 0.8614, 5.4218

4

6 28

47. 4 sin3 x + 2 sin2 x - 2 sin x - 1 = 0 Graph y = 4 sinax + 2 sin2x - 2 sin x - 1.

o ~ 6.28

49.

cos x cot x

1 - sin x - 3

Graph

y

=

(1

-

COS X

sin x) tan

x

-3.

3

The solutions are approximately x ~ 0.5236, x ~- 2.6180

By altering the y-range to Ymin = -.5 and Ymax = .5, you see that there are 6 solutions: 0.7854, 2.3562, 3.6652, 3.9270, 5.4978, 5.7596'. 51. xcosx- 1 =0

x ~ 4.9172

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

53. secZx + 0.5 tanx - 1 = 0 Graph Yl - (cos x)2 + 0.5 tan x - 1.

lO

0

6.28

-2

The x-intercepts occur at x = 0, x ~ 2.6779, x ~ 3.1416 and x ~ 5.8195.

55. 12sinzx- 13sinx+3 =0 Graph Yl = 12 sin:~ x - 13 sin x + 3.

40

-4

The x-intercepts occur at x ~ 0.3398, x ~ 0.8481, x ~" 2.2935, and x ~-- 2.8018.

57. 3tan2x+5tanx-4=0, -~,

tan x =

- 5 + ./25 - 4(- 4)(3) 2(3)

x ~ - 1.154, 0.535

59. 4cos2x-2sinx+l=0,[-~,~l

4(1 -sin2x)-2sinx+ 1 =0

-4sinZx-2sinx+5=0

sin x =

2

+ ,//4- 4(-4)(5) 2(- 4)

2 _+ ./g~

-8

--

-4

x ~ 1.110

61. (a) x 0

1

2

34

5

6

f(x) Undef. 0.83 -1.36 -2.93 -4.46 -6.34 - 13.02

The zero is in the interval (1, 2) since f changes signs in the interval.

(b) 2

The interval is the same as part (a). (c) 1.3065

~3. (a) x 0 1

2

3

4

5

6

f(x) -1 1.39 1.65 -0.70 - 1.94

- 1.48

The zeros are in the intervals (0, 1) and (2, 3) sincefchanges signs in these intervals.

8

The intervals are the same as part (a). (c) 0.4271, 2.7145

65. (a) f(x) = sin x + cos x

2

(b) cosx-sinx=O cos x = sin x sin x 1- cos X tan x = I ?r 57r x-4,4

f =sin~+cos4- 2 2

2 2 Therefore, the maximum point in the interval [0, 2"n')is (,n'/4, ~ and the minimum point is (5"n'/4, --,/~).

67. (a) f(x) = x sin 2x Maximum: (3.989, 3.958) Minimum: (5.543,-5.520)

(b) 2x cos 2x + sin 2x = 0 y = 2x cos 2x + sin 2x

lO

6 28 0 ~~~ 6.28

5 solutions: 0, 1.014, 2.457, 3.989, 5.543. The fourth and fifth correspond to the maximum and minimum found in part (a).

262 PART I: Solutions to Odd-Numbered Exercises and Practice Tekts

69. f(x) = tan 4 tan 0 = 0~ but 0 is not positive. By graphing y = tan-4- - x, you see that the smallest positive fixed point is x = 1.

71. f(x) = cos1-

x

(a) The domain off(x) is all real numbers except 0. (b) The graph has y-axis symmetry and a horizontal asymptote at y = 1.

(c) As x ----> 0, f(x) oscillates between - 1 and 1.

(d) There are an infinite number of solutions in the interval [- 1, 1 ].

(e) The greatest solution appears to occur at x -~- 0.6366.

73.

1 y = -j~(cos 8t - 3 sin 8t)

~2(cos 8t - 3 sin 8t) = 0

cos 8t -- 3 sin 8t

3-1 = tan 8t 8t = 0.32175 + n'n-

t = 0.04 + 8-In the interval 0 < t _< 1, t = 0.04, 0.43, and 0.83.

77. Range = 1000 yards = 3000 feet

vo = 1200 feet per second r = ~ Vo2 sin 20

3000 = (12oo)2 sin 20

sin 20 = 0.066667 20 ~-- 3.8226? 0 ~ 1.9113?

75. D=31 sm(3--~t - 1,4)

40

o I~1365 -40

D > 20? for 123 < t < 223 days

79. Yl = 1.56e-?'22t COS 4.9t intersects Y2 = -- 1 at t ~ 1.96

The displacement does not exceed one inch from equilibrium after t= 1.96 seconds.

Sl. f(x)= 3 sin(O.6x- 2) (a) Zero: sin(O.6x- 2) = 0 0.6x- 2 = 0 0.6x= 2 2 10 0.6 3 (c) -0.45xz + 5.52x - 13.70 = 0

X= - 5.52 + ./(5.52)2 - 4(-0.45)(- 13.70) 2(- 0.45)

x = 3.46, 8.81 The zero of g on [0, 6] is 3.46. The zero is close to the zero ~ ~ 3.33 off.

(b) g(x) = -0.45x2 + 5.52x - 13.70

5

For 3.5 < x < 6 the approximation appears to be good.

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

83. False. There might not be

periodicity, as in the equation sin(x2) = 0

2.164 radians

0.007 radians

89.

tan

30?-14 --~

x

x

=

tan1340?14=

--~

=

24.249

3

91. sin 40?=1x~6 ==~ x =sin1640? 24.892

95, f(x) = ~cot -

y

Section 5.4 Sum and Difference Formulas

[] You should memorize the sum and difference formulas. sin(u + v) = sin u cos v + cos u sin v cos(u + v) = cos u cos v 7- sin u sin v tan u + tan v tan(u + v) = 17- tan u tan v

[] You should be able to use these formulas to find the values of the trigonometric functions of angles whose sums or differences are special angles.

[] You should be able to use these formulas to solve trigonometric equations.

Solutions to Odd-Numbered Exercises 1, (a) cos + = cos -~ cos ~ - sin ~ sin ~-

(b) co"rsr ~+co,rsr -./~~=+T1 s

-,J'~" + 1

2

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