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  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Key Terms
    6. Key Concepts
    7. Review Exercises
    8. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  8. 7 The Unit Circle: Sine and Cosine Functions
    1. Introduction to The Unit Circle: Sine and Cosine Functions
    2. 7.1 Angles
    3. 7.2 Right Triangle Trigonometry
    4. 7.3 Unit Circle
    5. 7.4 The Other Trigonometric Functions
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  9. 8 Periodic Functions
    1. Introduction to Periodic Functions
    2. 8.1 Graphs of the Sine and Cosine Functions
    3. 8.2 Graphs of the Other Trigonometric Functions
    4. 8.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  10. 9 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 9.1 Solving Trigonometric Equations with Identities
    3. 9.2 Sum and Difference Identities
    4. 9.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 9.4 Sum-to-Product and Product-to-Sum Formulas
    6. 9.5 Solving Trigonometric Equations
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  11. 10 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 10.1 Non-right Triangles: Law of Sines
    3. 10.2 Non-right Triangles: Law of Cosines
    4. 10.3 Polar Coordinates
    5. 10.4 Polar Coordinates: Graphs
    6. 10.5 Polar Form of Complex Numbers
    7. 10.6 Parametric Equations
    8. 10.7 Parametric Equations: Graphs
    9. 10.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  12. 11 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 11.1 Systems of Linear Equations: Two Variables
    3. 11.2 Systems of Linear Equations: Three Variables
    4. 11.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 11.4 Partial Fractions
    6. 11.5 Matrices and Matrix Operations
    7. 11.6 Solving Systems with Gaussian Elimination
    8. 11.7 Solving Systems with Inverses
    9. 11.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  13. 12 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 12.1 The Ellipse
    3. 12.2 The Hyperbola
    4. 12.3 The Parabola
    5. 12.4 Rotation of Axes
    6. 12.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  14. 13 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 13.1 Sequences and Their Notations
    3. 13.2 Arithmetic Sequences
    4. 13.3 Geometric Sequences
    5. 13.4 Series and Their Notations
    6. 13.5 Counting Principles
    7. 13.6 Binomial Theorem
    8. 13.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  15. A | Proofs, Identities, and Toolkit Functions
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index

Try It

9.1 Solving Trigonometric Equations with Identities

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,G, F,G, and H, H, are even.

This is because 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 ) 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 ) 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 ) ), π 3 1 3 ( sin 1 ( 9 10 ) ), 2π 3 + 1 3 ( sin 1 ( 9 10 ) ),π 1 3 ( sin 1 ( 9 10 ) ), 4π 3 + 1 3 ( sin 1 ( 9 10 ) ), 5π 3 1 3 ( sin 1 ( 9 10 ) ) 1 3 ( sin 1 ( 9 10 ) ), π 3 1 3 ( sin 1 ( 9 10 ) ), 2π 3 + 1 3 ( sin 1 ( 9 10 ) ),π 1 3 ( sin 1 ( 9 10 ) ), 4π 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 ) ),2π cos 1 ( 1 3 ( 1 7 ) ) cos 1 ( 1 3 ( 1 7 ) ),2π cos 1 ( 1 3 ( 1 7 ) )

47.

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 ) ) 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 ) )

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

63.

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

65.

0, π 2 ,π, 3π 2 0, π 2 ,π, 3π 2

67.

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

69.

There are no solutions.

71.

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