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9.1 Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions

1.
cscθcosθtanθ = ( 1 sinθ )cosθ( sinθ cosθ ) = cosθ sinθ ( sinθ cosθ ) = sinθcosθ sinθcosθ = 1 cscθcosθtanθ = ( 1 sinθ )cosθ( sinθ cosθ ) = cosθ sinθ ( sinθ cosθ ) = sinθcosθ sinθcosθ = 1
2.
cotθ cscθ = cosθ sinθ 1 sinθ = cosθ sinθ sinθ 1 = cosθ cotθ cscθ = cosθ sinθ 1 sinθ = cosθ sinθ sinθ 1 = cosθ
3.

sin 2 θ1 tanθsinθtanθ = (sinθ+1)(sinθ1) tanθ(sinθ1) = sinθ+1 tanθ sin 2 θ1 tanθsinθtanθ = (sinθ+1)(sinθ1) tanθ(sinθ1) = sinθ+1 tanθ

4.

This is a difference of squares formula: 259 sin 2 θ=(53sinθ)(5+3sinθ). 259 sin 2 θ=(53sinθ)(5+3sinθ).

5.
cosθ 1+sinθ ( 1sinθ 1sinθ ) = cosθ(1sinθ) 1 sin 2 θ = cosθ(1sinθ) cos 2 θ = 1sinθ cosθ cosθ 1+sinθ ( 1sinθ 1sinθ ) = cosθ(1sinθ) 1 sin 2 θ = cosθ(1sinθ) cos 2 θ = 1sinθ cosθ

9.2 Sum and Difference Identities

1.

2 + 6 4 2 + 6 4

2.

2 6 4 2 6 4

3.

1 3 1+ 3 1 3 1+ 3

4.

cos( 5π 14 ) cos( 5π 14 )

5.
tan(πθ) = tan(π)tanθ 1+tan(π)tanθ = 0tanθ 1+0tanθ = tanθ tan(πθ) = tan(π)tanθ 1+tan(π)tanθ = 0tanθ 1+0tanθ = tanθ

9.3 Double-Angle, Half-Angle, and Reduction Formulas

1.

cos( 2α )= 7 32 cos( 2α )= 7 32

2.

cos 4 θ sin 4 θ=( cos 2 θ+ sin 2 θ )( cos 2 θ sin 2 θ )=cos( 2θ ) cos 4 θ sin 4 θ=( cos 2 θ+ sin 2 θ )( cos 2 θ sin 2 θ )=cos( 2θ )

3.

cos( 2θ )cosθ=( cos 2 θ sin 2 θ )cosθ= cos 3 θcosθ sin 2 θ cos( 2θ )cosθ=( cos 2 θ sin 2 θ )cosθ= cos 3 θcosθ sin 2 θ

4.

10 cos 4 x = 10 ( cos 2 x ) 2 = 10 [ 1+cos(2x) 2 ] 2 Substitute reduction formula for cos 2 x. = 10 4 [1+2cos(2x)+ cos 2 (2x)] = 10 4 + 10 2 cos(2x)+ 10 4 ( 1+cos2(2x) 2 ) Substitute reduction formula for cos 2 x. = 10 4 + 10 2 cos(2x)+ 10 8 + 10 8 cos(4x) = 30 8 +5cos(2x)+ 10 8 cos(4x) = 15 4 +5cos(2x)+ 5 4 cos(4x) 10 cos 4 x = 10 ( cos 2 x ) 2 = 10 [ 1+cos(2x) 2 ] 2 Substitute reduction formula for cos 2 x. = 10 4 [1+2cos(2x)+ cos 2 (2x)] = 10 4 + 10 2 cos(2x)+ 10 4 ( 1+cos2(2x) 2 ) Substitute reduction formula for cos 2 x. = 10 4 + 10 2 cos(2x)+ 10 8 + 10 8 cos(4x) = 30 8 +5cos(2x)+ 10 8 cos(4x) = 15 4 +5cos(2x)+ 5 4 cos(4x)

5.

2 5 2 5

9.4 Sum-to-Product and Product-to-Sum Formulas

1.

1 2 ( cos6θ+cos2θ ) 1 2 ( cos6θ+cos2θ )

2.

1 2 ( sin2x+sin2y ) 1 2 ( sin2x+sin2y )

3.

2 3 4 2 3 4

4.

2sin( 2θ )cos( θ ) 2sin( 2θ )cos( θ )

5.

tanθcotθ cos 2 θ = ( sinθ cosθ )( cosθ sinθ ) cos 2 θ = 1 cos 2 θ = sin 2 θ tanθcotθ cos 2 θ = ( sinθ cosθ )( cosθ sinθ ) cos 2 θ = 1 cos 2 θ = sin 2 θ

9.5 Solving Trigonometric Equations

1.

x= 7π 6 , 11π 6 x= 7π 6 , 11π 6

2.

π 3 ±πk π 3 ±πk

3.

θ1.7722±2πk θ1.7722±2πk and θ4.5110±2πk θ4.5110±2πk

4.

cosθ=1,θ=π cosθ=1,θ=π

5.

π 2 , 2π 3 , 4π 3 , 3π 2 π 2 , 2π 3 , 4π 3 , 3π 2

9.1 Section Exercises

1.

All three functions, F F,GG, and H H, are even.

This is because F( x )=sin( x )sin( x )=( sinx )( sinx )= sin 2 x=F( x ) F( x )=sin( x )sin( x )=( sinx )( sinx )= sin 2 x=F( x ),G( x )=cos( x )cos( x )=cosxcosx= cos 2 x=G( x ) G( x )=cos( x )cos( x )=cosxcosx= cos 2 x=G( x ) and H( x )=tan( x )tan( x )=( tanx )( tanx )= tan 2 x=H( x ). H( x )=tan( x )tan( x )=( tanx )( tanx )= tan 2 x=H( x ).

3.

When cost=0, cost=0, then sect= 1 0 , sect= 1 0 , which is undefined.

5.

sinx sinx

7.

secx secx

9.

csct csct

11.

−1 −1

13.

sec 2 x sec 2 x

15.

sin 2 x+1 sin 2 x+1

17.

1 sinx 1 sinx

19.

1 cotx 1 cotx

21.

tanx tanx

23.

4secxtanx 4secxtanx

25.

± 1 cot 2 x +1 ± 1 cot 2 x +1

27.

± 1 sin 2 x sinx ± 1 sin 2 x sinx

29.

Answers will vary. Sample proof:

cosx cos 3 x = cosx(1 cos 2 x) = cosx sin 2 x cosx cos 3 x = cosx(1 cos 2 x) = cosx sin 2 x

31.

Answers will vary. Sample proof:
1+ sin 2 x cos 2 x = 1 cos 2 x + sin 2 x cos 2 x = sec 2 x+ tan 2 x= tan 2 x+1+ tan 2 x=1+2 tan 2 x 1+ sin 2 x cos 2 x = 1 cos 2 x + sin 2 x cos 2 x = sec 2 x+ tan 2 x= tan 2 x+1+ tan 2 x=1+2 tan 2 x

33.

Answers will vary. Sample proof:
cos 2 x tan 2 x=1 sin 2 x( sec 2 x1 )=1 sin 2 x sec 2 x+1=2 sin 2 x sec 2 x cos 2 x tan 2 x=1 sin 2 x( sec 2 x1 )=1 sin 2 x sec 2 x+1=2 sin 2 x sec 2 x

35.

False

37.

False

39.

Proved with negative and Pythagorean identities

41.

True 3 sin 2 θ+4 cos 2 θ=3 sin 2 θ+3 cos 2 θ+ cos 2 θ=3( sin 2 θ+ cos 2 θ )+ cos 2 θ=3+ cos 2 θ 3 sin 2 θ+4 cos 2 θ=3 sin 2 θ+3 cos 2 θ+ cos 2 θ=3( sin 2 θ+ cos 2 θ )+ cos 2 θ=3+ cos 2 θ

9.2 Section Exercises

1.

The cofunction identities apply to complementary angles. Viewing the two acute angles of a right triangle, if one of those angles measures x, x, the second angle measures π 2 x. π 2 x. Then sinx=cos( π 2 x ). sinx=cos( π 2 x ). The same holds for the other cofunction identities. The key is that the angles are complementary.

3.

sin( x )=sinx, sin( x )=sinx, so sinx sinx is odd. cos( x )=cos( 0x )=cosx, cos( x )=cos( 0x )=cosx, so cosx cosx is even.

5.

2 + 6 4 2 + 6 4

7.

6 2 4 6 2 4

9.

2 3 2 3

11.

2 2 sinx 2 2 cosx 2 2 sinx 2 2 cosx

13.

1 2 cosx 3 2 sinx 1 2 cosx 3 2 sinx

15.

cscθ cscθ

17.

cotx cotx

19.

tan( x 10 ) tan( x 10 )

21.

sin(ab) = ( 4 5 )( 1 3 )( 3 5 )( 2 2 3 ) = 46 2 15 cos(a+b) = ( 3 5 )( 1 3 )( 4 5 )( 2 2 3 ) = 38 2 15 sin(ab) = ( 4 5 )( 1 3 )( 3 5 )( 2 2 3 ) = 46 2 15 cos(a+b) = ( 3 5 )( 1 3 )( 4 5 )( 2 2 3 ) = 38 2 15

23.

2 6 4 2 6 4

25.

sinx sinx

Graph of y=sin(x) from -2pi to 2pi.
27.

cot( π 6 x ) cot( π 6 x )

Graph of y=cot(pi/6 - x) from -2pi to pi - in comparison to the usual y=cot(x) graph, this one is reflected across the x-axis and shifted by pi/6.
29.

cot( π 4 +x ) cot( π 4 +x )

Graph of y=cot(pi/4 + x) - in comparison to the usual y=cot(x) graph, this one is shifted by pi/4.
31.

sinx 2 + cosx 2 sinx 2 + cosx 2

Graph of y = sin(x) / rad2 + cos(x) / rad2 - it looks like the sin curve shifted by pi/4.
33.

They are the same.

35.

They are the different, try g( x )=sin( 9x )cos( 3x )sin( 6x ). g( x )=sin( 9x )cos( 3x )sin( 6x ).

37.

They are the same.

39.

They are the different, try g( θ )= 2tanθ 1 tan 2 θ . g( θ )= 2tanθ 1 tan 2 θ .

41.

They are different, try g( x )= tanxtan( 2x ) 1+tanxtan( 2x ) . g( x )= tanxtan( 2x ) 1+tanxtan( 2x ) .

43.

3 1 2 2 ,or 0.2588 3 1 2 2 ,or 0.2588

45.

1+ 3 2 2 , 1+ 3 2 2 , or 0.9659

47.

tan( x+ π 4 ) = tanx+tan( π 4 ) 1tanxtan( π 4 ) = tanx+1 1tanx(1) = tanx+1 1tanx tan( x+ π 4 ) = tanx+tan( π 4 ) 1tanxtan( π 4 ) = tanx+1 1tanx(1) = tanx+1 1tanx

49.

cos(a+b) cosacosb = cosacosb cosacosb sinasinb cosacosb = 1tanatanb cos(a+b) cosacosb = cosacosb cosacosb sinasinb cosacosb = 1tanatanb

51.

cos(x+h)cosx h = cosxcoshsinxsinhcosx h = cosx(cosh1)sinxsinh h = cosx cosh1 h sinx sinh h cos(x+h)cosx h = cosxcoshsinxsinhcosx h = cosx(cosh1)sinxsinh h = cosx cosh1 h sinx sinh h

53.

True

55.

True. Note that sin( α+β )=sin( πγ ) sin( α+β )=sin( πγ ) and expand the right hand side.

9.3 Section Exercises

1.

Use the Pythagorean identities and isolate the squared term.

3.

1cosx sinx , sinx 1+cosx , 1cosx sinx , sinx 1+cosx , multiplying the top and bottom by 1cosx 1cosx and 1+cosx , 1+cosx , respectively.

5.

a) 3 7 32 3 7 32 b) 31 32 31 32 c) 3 7 31 3 7 31

7.

a) 3 2 3 2 b) 1 2 1 2 c) 3 3

9.

cosθ= 2 5 5 ,sinθ= 5 5 ,tanθ= 1 2 ,cscθ= 5 ,secθ= 5 2 ,cotθ=2 cosθ= 2 5 5 ,sinθ= 5 5 ,tanθ= 1 2 ,cscθ= 5 ,secθ= 5 2 ,cotθ=2

11.

2sin( π 2 ) 2sin( π 2 )

13.

2 2 2 2 2 2

15.

2 3 2 2 3 2

17.

2+ 3 2+ 3

19.

1 2 1 2

21.

a) 3 13 13 3 13 13 b) 2 13 13 2 13 13 c) 3 2 3 2

23.

a) 10 4 10 4 b) 6 4 6 4 c) 15 3 15 3

25.

120 169 , 119 169 , 120 119 120 169 , 119 169 , 120 119

27.

2 13 13 , 3 13 13 , 2 3 2 13 13 , 3 13 13 , 2 3

29.

cos(74°) cos(74°)

31.

cos(18x) cos(18x)

33.

3sin(10x) 3sin(10x)

35.

2sin( x )cos( x )=2(sin( x )cos( x ))=sin( 2x ) 2sin( x )cos( x )=2(sin( x )cos( x ))=sin( 2x )

37.

sin(2θ) 1+cos(2θ) tan 2 θ = 2sin(θ)cos(θ) 1+ cos 2 θ sin 2 θ tan 2 θ= 2sin(θ)cos(θ) 2 cos 2 θ tan 2 θ = sin(θ) cosθ tan 2 θ= cot(θ) tan 2 θ = tan3 θ sin(2θ) 1+cos(2θ) tan 2 θ = 2sin(θ)cos(θ) 1+ cos 2 θ sin 2 θ tan 2 θ= 2sin(θ)cos(θ) 2 cos 2 θ tan 2 θ = sin(θ) cosθ tan 2 θ= cot(θ) tan 2 θ = tan3 θ

39.

1+cos(12x) 2 1+cos(12x) 2

41.

3+cos(12x)4cos(6x) 8 3+cos(12x)4cos(6x) 8

43.

2+cos(2x)2cos(4x)cos(6x) 32 2+cos(2x)2cos(4x)cos(6x) 32

45.

3+cos(4x)4cos(2x) 3+cos(4x)+4cos(2x) 3+cos(4x)4cos(2x) 3+cos(4x)+4cos(2x)

47.

1cos(4x) 8 1cos(4x) 8

49.

3+cos(4x)4cos(2x) 4(cos(2x)+1) 3+cos(4x)4cos(2x) 4(cos(2x)+1)

51.

( 1+cos( 4x ) )sinx 2 ( 1+cos( 4x ) )sinx 2

53.

4sinxcosx( cos 2 x sin 2 x ) 4sinxcosx( cos 2 x sin 2 x )

55.

2tanx 1+ tan 2 x = 2sinx cosx 1+ sin 2 x cos 2 x = 2sinx cosx cos 2 x+ sin 2 x cos 2 x = 2sinx cosx . cos 2 x 1 =2sinxcosx=sin(2x) 2tanx 1+ tan 2 x = 2sinx cosx 1+ sin 2 x cos 2 x = 2sinx cosx cos 2 x+ sin 2 x cos 2 x = 2sinx cosx . cos 2 x 1 =2sinxcosx=sin(2x)

57.

2sinxcosx 2 cos 2 x1 = sin(2x) cos(2x) =tan(2x) 2sinxcosx 2 cos 2 x1 = sin(2x) cos(2x) =tan(2x)

59.

sin(x+2x) = sinxcos(2x)+sin(2x)cosx = sinx( cos 2 x sin 2 x )+2sinxcosxcosx = sinx cos 2 x sin 3 x+2sinx cos 2 x = 3sinx cos 2 x sin 3 x sin(x+2x) = sinxcos(2x)+sin(2x)cosx = sinx( cos 2 x sin 2 x )+2sinxcosxcosx = sinx cos 2 x sin 3 x+2sinx cos 2 x = 3sinx cos 2 x sin 3 x

61.

1+cos(2t) sin(2t)cost = 1+2 cos 2 t1 2sintcostcost = 2 cos 2 t cost(2sint1) = 2cost 2sint1 1+cos(2t) sin(2t)cost = 1+2 cos 2 t1 2sintcostcost = 2 cos 2 t cost(2sint1) = 2cost 2sint1

63.

( cos 2 (4x) sin 2 (4x)sin(8x))( cos 2 (4x) sin 2 (4x)+sin(8x) ) = = (cos(8x)sin(8x))(cos(8x)+sin(8x)) = cos 2 (8x) sin 2 (8x) = cos(16x) ( cos 2 (4x) sin 2 (4x)sin(8x))( cos 2 (4x) sin 2 (4x)+sin(8x) ) = = (cos(8x)sin(8x))(cos(8x)+sin(8x)) = cos 2 (8x) sin 2 (8x) = cos(16x)

9.4 Section Exercises

1.

Substitute α α into cosine and β β into sine and evaluate.

3.

Answers will vary. There are some equations that involve a sum of two trig expressions where when converted to a product are easier to solve. For example: sin(3x)+sinx cosx =1. sin(3x)+sinx cosx =1. When converting the numerator to a product the equation becomes: 2sin(2x)cosx cosx =1 2sin(2x)cosx cosx =1

5.

8( cos( 5x )cos( 27x ) ) 8( cos( 5x )cos( 27x ) )

7.

sin( 2x )+sin( 8x ) sin( 2x )+sin( 8x )

9.

1 2 ( cos( 6x )cos( 4x ) ) 1 2 ( cos( 6x )cos( 4x ) )

11.

2cos( 5t )cost 2cos( 5t )cost

13.

2cos( 7x ) 2cos( 7x )

15.

2cos( 6x )cos( 3x ) 2cos( 6x )cos( 3x )

17.

1 4 ( 1+ 3 ) 1 4 ( 1+ 3 )

19.

1 4 ( 3 2 ) 1 4 ( 3 2 )

21.

1 4 ( 3 1 ) 1 4 ( 3 1 )

23.

cos( 80° )cos( 120° ) cos( 80° )cos( 120° )

25.

1 2 (sin(221°)+sin(205°)) 1 2 (sin(221°)+sin(205°))

27.

2 cos( 31° ) 2 cos( 31° )

29.

2cos(66.5°)sin(34.5°) 2cos(66.5°)sin(34.5°)

31.

2sin( −1.5° )cos( 0.5° ) 2sin( −1.5° )cos( 0.5° )

33.

2sin(7x)2sinx=2sin(4x+3x)2sin(4x3x)= 2(sin(4x)cos(3x)+sin(3x)cos(4x))2(sin(4x)cos(3x)sin(3x)cos(4x))= 2sin(4x)cos(3x)+2sin(3x)cos(4x))2sin(4x)cos(3x)+2sin(3x)cos(4x))= 4sin(3x)cos(4x) 2sin(7x)2sinx=2sin(4x+3x)2sin(4x3x)= 2(sin(4x)cos(3x)+sin(3x)cos(4x))2(sin(4x)cos(3x)sin(3x)cos(4x))= 2sin(4x)cos(3x)+2sin(3x)cos(4x))2sin(4x)cos(3x)+2sin(3x)cos(4x))= 4sin(3x)cos(4x)

35.

sinx+sin(3x) = 2sin( 4x 2 )cos( 2x 2 )= 2sin(2x)cosx = 2(2sinxcosx)cosx= 4sinx cos 2 x sinx+sin(3x) = 2sin( 4x 2 )cos( 2x 2 )= 2sin(2x)cosx = 2(2sinxcosx)cosx= 4sinx cos 2 x

37.

2tanxcos( 3x )= 2sinxcos(3x) cosx = 2(.5(sin(4x)sin(2x))) cosx = 1 cosx ( sin(4x)sin(2x) )=secx( sin( 4x )sin( 2x ) ) 2tanxcos( 3x )= 2sinxcos(3x) cosx = 2(.5(sin(4x)sin(2x))) cosx = 1 cosx ( sin(4x)sin(2x) )=secx( sin( 4x )sin( 2x ) )

39.

2cos(35°)cos(23°),1.5081 2cos(35°)cos(23°),1.5081

41.

2sin(33°)sin(11°),0.2078 2sin(33°)sin(11°),0.2078

43.

1 2 (cos(99°)cos(71°)),−0.2410 1 2 (cos(99°)cos(71°)),−0.2410

45.

It is an identity.

47.

It is not an identity, but 2 cos 3 x 2 cos 3 x is.

49.

tan( 3t ) tan( 3t )

51.

2cos( 2x ) 2cos( 2x )

53.

sin(14x) sin(14x)

55.

Start with cosx+cosy. cosx+cosy. Make a substitution and let x=α+β x=α+β and let y=αβ, y=αβ, so cosx+cosy cosx+cosy becomes cos(α+β)+cos(αβ)=cosαcosβsinαsinβ+cosαcosβ+sinαsinβ= 2cosαcosβ cos(α+β)+cos(αβ)=cosαcosβsinαsinβ+cosαcosβ+sinαsinβ= 2cosαcosβ

Since x=α+β x=α+β and y=αβ, y=αβ, we can solve for α α and β β in terms of x and y and substitute in for 2cosαcosβ 2cosαcosβ and get 2cos( x+y 2 )cos( xy 2 ). 2cos( x+y 2 )cos( xy 2 ).

57.

cos( 3x )+cosx cos( 3x )cosx = 2cos( 2x )cosx 2sin( 2x )sinx =cot( 2x )cotx cos( 3x )+cosx cos( 3x )cosx = 2cos( 2x )cosx 2sin( 2x )sinx =cot( 2x )cotx

59.

cos(2y)cos(4y) sin(2y)+sin(4y) = 2sin(3y)sin(y) 2sin(3y)cosy = 2sin(3y)sin(y) 2sin(3y)cosy = tany cos(2y)cos(4y) sin(2y)+sin(4y) = 2sin(3y)sin(y) 2sin(3y)cosy = 2sin(3y)sin(y) 2sin(3y)cosy = tany

61.

cosxcos( 3x )=2sin(2x)sin(x)= 2(2sinxcosx)sinx=4 sin 2 xcosx cosxcos( 3x )=2sin(2x)sin(x)= 2(2sinxcosx)sinx=4 sin 2 xcosx

63.

tan( π 4 t )= tan( π 4 )tant 1+tan( π 4 )tan(t) = 1tant 1+tant tan( π 4 t )= tan( π 4 )tant 1+tan( π 4 )tan(t) = 1tant 1+tant

9.5 Section Exercises

1.

There will not always be solutions to trigonometric function equations. For a basic example, cos(x)=−5. cos(x)=−5.

3.

If the sine or cosine function has a coefficient of one, isolate the term on one side of the equals sign. If the number it is set equal to has an absolute value less than or equal to one, the equation has solutions, otherwise it does not. If the sine or cosine does not have a coefficient equal to one, still isolate the term but then divide both sides of the equation by the leading coefficient. Then, if the number it is set equal to has an absolute value greater than one, the equation has no solution.

5.

π 3 , 2π 3 π 3 , 2π 3

7.

3π 4 , 5π 4 3π 4 , 5π 4

9.

π 4 , 5π 4 π 4 , 5π 4

11.

π 4 , 3π 4 , 5π 4 , 7π 4 π 4 , 3π 4 , 5π 4 , 7π 4

13.

π 4 , 7π 4 π 4 , 7π 4

15.

7π 6 , 11π 6 7π 6 , 11π 6

17.

π 18 , 5π 18 , 13π 18 , 17π 18 , 25π 18 , 29π 18 π 18 , 5π 18 , 13π 18 , 17π 18 , 25π 18 , 29π 18

19.

3π 12 , 5π 12 , 11π 12 , 13π 12 , 19π 12 , 21π 12 3π 12 , 5π 12 , 11π 12 , 13π 12 , 19π 12 , 21π 12

21.

1 6 , 5 6 , 13 6 , 17 6 , 25 6 , 29 6 , 37 6 1 6 , 5 6 , 13 6 , 17 6 , 25 6 , 29 6 , 37 6

23.

0, π 3 ,π, 5π 3 0, π 3 ,π, 5π 3

25.

π 3 ,π, 5π 3 π 3 ,π, 5π 3

27.

π 3 , 3π 2 , 5π 3 π 3 , 3π 2 , 5π 3

29.

0,π 0,π

31.

π sin 1 ( 1 4 ), 7π 6 , 11π 6 ,2π+ sin 1 ( 1 4 ) π sin 1 ( 1 4 ), 7π 6 , 11π 6 ,2π+ sin 1 ( 1 4 )

33.

1 3 ( sin 1 ( 9 10 ) ) 1 3 ( sin 1 ( 9 10 ) ), π 3 1 3 ( sin 1 ( 9 10 ) ) π 3 1 3 ( sin 1 ( 9 10 ) ), 2π 3 + 1 3 ( sin 1 ( 9 10 ) ) 2π 3 + 1 3 ( sin 1 ( 9 10 ) ), π 1 3 ( sin 1 ( 9 10 ) )π 1 3 ( sin 1 ( 9 10 ) ), 4π 3 + 1 3 ( sin 1 ( 9 10 ) ) 4π 3 + 1 3 ( sin 1 ( 9 10 ) ), 5π 3 1 3 ( sin 1 ( 9 10 ) ) 5π 3 1 3 ( sin 1 ( 9 10 ) )

35.

0 0

37.

π 6 , 5π 6 , 7π 6 , 11π 6 π 6 , 5π 6 , 7π 6 , 11π 6

39.

3π 2 , π 6 , 5π 6 3π 2 , π 6 , 5π 6

41.

0, π 3 ,π, 4π 3 0, π 3 ,π, 4π 3

43.

There are no solutions.

45.

cos 1 ( 1 3 ( 1 7 ) ) cos 1 ( 1 3 ( 1 7 ) ), 2π cos 1 ( 1 3 ( 1 7 ) ) 2π cos 1 ( 1 3 ( 1 7 ) )

47.

tan 1 ( 1 2 ( 29 5 ) ) tan 1 ( 1 2 ( 29 5 ) ), π+ tan 1 ( 1 2 ( 29 5 ) )π+ tan 1 ( 1 2 ( 29 5 ) ), π+ tan 1 ( 1 2 ( 29 5 ) )π+ tan 1 ( 1 2 ( 29 5 ) ), 2π+ tan 1 ( 1 2 ( 29 5 ) ) 2π+ tan 1 ( 1 2 ( 29 5 ) )

49.

There are no solutions.

51.

There are no solutions.

53.

0, 2π 3 , 4π 3 0, 2π 3 , 4π 3

55.

π 4 , 3π 4 , 5π 4 , 7π 4 π 4 , 3π 4 , 5π 4 , 7π 4

57.

sin 1 ( 3 5 ), π 2 ,π sin 1 ( 3 5 ), 3π 2 sin 1 ( 3 5 ), π 2 ,π sin 1 ( 3 5 ), 3π 2

59.

cos 1 ( 1 4 ),2π cos 1 ( 1 4 ) cos 1 ( 1 4 ),2π cos 1 ( 1 4 )

61.

π 3 π 3 , cos 1 ( 3 4 ) cos 1 ( 3 4 ), 2π cos 1 ( 3 4 )2π cos 1 ( 3 4 ), 5π 3 5π 3

63.

cos 1 ( 3 4 ) cos 1 ( 3 4 ), cos 1 ( 2 3 ) cos 1 ( 2 3 ), 2π cos 1 ( 2 3 )2π cos 1 ( 2 3 ), 2π cos 1 ( 3 4 ) 2π cos 1 ( 3 4 )

65.

0, π 2 ,π, 3π 2 0, π 2 ,π, 3π 2

67.

π 3 π 3 , cos −1 ( 1 4 ) cos −1 ( 1 4 ), 2π cos −1 ( 1 4 )2π cos −1 ( 1 4 ), 5π 3 5π 3

69.

There are no solutions.

71.

π+ tan −1 ( −2 ) π+ tan −1 ( −2 ), π+ tan −1 ( 3 2 )π+ tan −1 ( 3 2 ), 2π+ tan −1 ( −2 )2π+ tan −1 ( −2 ), 2π+ tan −1 ( 3 2 ) 2π+ tan −1 ( 3 2 )

73.

2πk+0.2734,2πk+2.8682 2πk+0.2734,2πk+2.8682

75.

πk0.3277 πk0.3277

77.

0.6694,1.8287,3.8110,4.9703 0.6694,1.8287,3.8110,4.9703

79.

1.0472,3.1416,5.2360 1.0472,3.1416,5.2360

81.

0.5326,1.7648,3.6742,4.9064 0.5326,1.7648,3.6742,4.9064

83.

sin 1 ( 1 4 ),π sin 1 ( 1 4 ), 3π 2 sin 1 ( 1 4 ),π sin 1 ( 1 4 ), 3π 2

85.

π 2 , 3π 2 π 2 , 3π 2

87.

There are no solutions.

89.

0, π 2 ,π, 3π 2 0, π 2 ,π, 3π 2

91.

There are no solutions.

93.

7.2 7.2

95.

5.7 5.7

97.

82.4 82.4

99.

31.0 31.0

101.

88.7 88.7

103.

59.0 59.0

105.

36.9 36.9

Review Exercises

1.

sin 1 ( 3 3 ) sin 1 ( 3 3 ), π sin 1 ( 3 3 )π sin 1 ( 3 3 ), π+ sin 1 ( 3 3 )π+ sin 1 ( 3 3 ), 2π sin 1 ( 3 3 ) 2π sin 1 ( 3 3 )

3.

7π 6 , 11π 6 7π 6 , 11π 6

5.

sin 1 ( 1 4 ),π sin 1 ( 1 4 ) sin 1 ( 1 4 ),π sin 1 ( 1 4 )

7.

1 1

9.

Yes

11.

2 3 2 3

13.

2 2 2 2

15.

cos(4x)cos(3x)cosx = cos(2x+2x)cos(x+2x)cosx = cos(2x)cos(2x)sin(2x)sin(2x)cosxcos(2x)cosx+sinxsin(2x)cosx = ( cos 2 x sin 2 x ) 2 4 cos 2 x sin 2 x cos 2 x( cos 2 x sin 2 x )+sinx(2)sinxcosxcosx = ( cos 2 x sin 2 x ) 2 4 cos 2 x sin 2 x cos 2 x( cos 2 x sin 2 x )+2 sin 2 x cos 2 x = cos 4 x2 cos 2 x sin 2 x+ sin 4 x4 cos 2 x sin 2 x cos 4 x+ cos 2 x sin 2 x+2 sin 2 x cos 2 x = sin 4 x4 cos 2 x sin 2 x+ cos 2 x sin 2 x = sin 2 x( sin 2 x+ cos 2 x )4 cos 2 x sin 2 x = sin 2 x4 cos 2 x sin 2 x cos(4x)cos(3x)cosx = cos(2x+2x)cos(x+2x)cosx = cos(2x)cos(2x)sin(2x)sin(2x)cosxcos(2x)cosx+sinxsin(2x)cosx = ( cos 2 x sin 2 x ) 2 4 cos 2 x sin 2 x cos 2 x( cos 2 x sin 2 x )+sinx(2)sinxcosxcosx = ( cos 2 x sin 2 x ) 2 4 cos 2 x sin 2 x cos 2 x( cos 2 x sin 2 x )+2 sin 2 x cos 2 x = cos 4 x2 cos 2 x sin 2 x+ sin 4 x4 cos 2 x sin 2 x cos 4 x+ cos 2 x sin 2 x+2 sin 2 x cos 2 x = sin 4 x4 cos 2 x sin 2 x+ cos 2 x sin 2 x = sin 2 x( sin 2 x+ cos 2 x )4 cos 2 x sin 2 x = sin 2 x4 cos 2 x sin 2 x

17.

tan( 5 8 x ) tan( 5 8 x )

19.

3 3 3 3

21.

24 25 , 7 25 , 24 7 24 25 , 7 25 , 24 7

23.

2( 2+ 2 ) 2( 2+ 2 )

25.

2 10 , 7 2 10 , 1 7 , 3 5 , 4 5 , 3 4 2 10 , 7 2 10 , 1 7 , 3 5 , 4 5 , 3 4

27.

cotxcos(2x) = cotx( 12 sin 2 x ) = cotx cosx sinx (2) sin 2 x = 2sinxcosx+cotx = sin(2x)+cotx cotxcos(2x) = cotx( 12 sin 2 x ) = cotx cosx sinx (2) sin 2 x = 2sinxcosx+cotx = sin(2x)+cotx

29.

10sinx5sin( 3x )+sin( 5x ) 8( cos( 2x )+1 ) 10sinx5sin( 3x )+sin( 5x ) 8( cos( 2x )+1 )

31.

3 2 3 2

33.

2 2 2 2

35.

1 2 ( sin(6x)+sin(12x) ) 1 2 ( sin(6x)+sin(12x) )

37.

2sin( 13 2 x )cos( 9 2 x ) 2sin( 13 2 x )cos( 9 2 x )

39.

3π 4 , 7π 4 3π 4 , 7π 4

41.

0, π 6 , 5π 6 ,π 0, π 6 , 5π 6 ,π

43.

3π 2 3π 2

45.

No solution

47.

0.2527,2.8889,4.7124 0.2527,2.8889,4.7124

49.

1.3694,1.9106,4.3726,4.9137 1.3694,1.9106,4.3726,4.9137

Practice Test

1.

1

3.

sec( θ ) sec( θ )

5.

2 6 4 2 6 4

7.

2 3 2 3

9.

1 2 cos θ + 3 2 sin θ 1 2 cos θ + 3 2 sin θ

11.

1cos( 64 ) 2 1cos( 64 ) 2

13.

0,π 0,π

15.

π2,3π2 π2,3π2

17.

2cos(3x)cos(5x)2cos(3x)cos(5x)

19.

4sin( 2θ )cos( 6θ ) 4sin( 2θ )cos( 6θ )

21.

x=cos–1 (15) x=cos–1 (15)

23.

π3 π3

25.

35 , 45 , 34 35 , 45 , 34

27.

tan3xtanxsec2x =tanx(tan2xsec2x) =tanx(tan2x(1+tan2x)) =tanx(tan2x1tan2x) =tanx=tan(x)=tan(x) tan3xtanxsec2x =tanx(tan2xsec2x) =tanx(tan2x(1+tan2x)) =tanx(tan2x1tan2x) =tanx=tan(x)=tan(x)

29.

sin(2x) sinx cos(2x) cosx = 2sinxcosx sinx 2cos2x1cosx = 2cosx2cosx+1cosx = 1cosx =secx=secx sin(2x) sinx cos(2x) cosx = 2sinxcosx sinx 2cos2x1cosx = 2cosx2cosx+1cosx = 1cosx =secx=secx

31.

Amplitude: 1414 , period: 160160 , frequency: 60 Hz

33.

Amplitude: 8, fast period: 15001500 , fast frequency: 500 Hz, slow period: 110110 , slow frequency: 10 Hz

35.

D(t)=20(0.9086)t cos(4πt) D(t)=20(0.9086)t cos(4πt) , 31 second

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