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Algebra and Trigonometry

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

Algebra and Trigonometry9.3 Double-Angle, Half-Angle, and Reduction Formulas
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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

Learning Objectives

In this section, you will:
  • Use double-angle formulas to find exact values.
  • Use double-angle formulas to verify identities.
  • Use reduction formulas to simplify an expression.
  • Use half-angle formulas to find exact values.
Picture of two bicycle ramps, one with a steep slope and one with a gentle slope.
Figure 1 Bicycle ramps for advanced riders have a steeper incline than those designed for novices.

Bicycle ramps made for competition (see Figure 1) must vary in height depending on the skill level of the competitors. For advanced competitors, the angle formed by the ramp and the ground should be θ θ such that tanθ= 5 3 . tanθ= 5 3 . The angle is divided in half for novices. What is the steepness of the ramp for novices? In this section, we will investigate three additional categories of identities that we can use to answer questions such as this one.

Using Double-Angle Formulas to Find Exact Values

In the previous section, we used addition and subtraction formulas for trigonometric functions. Now, we take another look at those same formulas. The double-angle formulas are a special case of the sum formulas, where α=β. α=β. Deriving the double-angle formula for sine begins with the sum formula,

sin( α+β )=sinαcosβ+cosαsinβ sin( α+β )=sinαcosβ+cosαsinβ

If we let α=β=θ, α=β=θ, then we have

sin(θ+θ) = sinθcosθ+cosθsinθ sin(2θ) = 2sinθcosθ sin(θ+θ) = sinθcosθ+cosθsinθ sin(2θ) = 2sinθcosθ

Deriving the double-angle for cosine gives us three options. First, starting from the sum formula, cos( α+β )=cosαcosβsinαsinβ, cos( α+β )=cosαcosβsinαsinβ, and letting α=β=θ, α=β=θ, we have

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

Using the Pythagorean properties, we can expand this double-angle formula for cosine and get two more variations. The first variation is:

cos(2θ) = cos 2 θ sin 2 θ = ( 1 sin 2 θ ) sin 2 θ = 12 sin 2 θ cos(2θ) = cos 2 θ sin 2 θ = ( 1 sin 2 θ ) sin 2 θ = 12 sin 2 θ

The second variation is:

cos(2θ) = cos 2 θ sin 2 θ = cos 2 θ( 1 cos 2 θ ) = 2 cos 2 θ1 cos(2θ) = cos 2 θ sin 2 θ = cos 2 θ( 1 cos 2 θ ) = 2 cos 2 θ1

Similarly, to derive the double-angle formula for tangent, replacing α=β=θ α=β=θ in the sum formula gives

tan(α+β) = tanα+tanβ 1tanαtanβ tan(θ+θ) = tanθ+tanθ 1tanθtanθ tan(2θ) = 2tanθ 1 tan 2 θ tan(α+β) = tanα+tanβ 1tanαtanβ tan(θ+θ) = tanθ+tanθ 1tanθtanθ tan(2θ) = 2tanθ 1 tan 2 θ

Double-Angle Formulas

The double-angle formulas are summarized as follows:

sin(2θ) = 2sinθcosθ sin(2θ) = 2sinθcosθ

cos(2θ) = cos 2 θ sin 2 θ = 12 sin 2 θ = 2 cos 2 θ1 cos(2θ) = cos 2 θ sin 2 θ = 12 sin 2 θ = 2 cos 2 θ1

tan(2θ) = 2tanθ 1 tan 2 θ tan(2θ) = 2tanθ 1 tan 2 θ

How To

Given the tangent of an angle and the quadrant in which it is located, use the double-angle formulas to find the exact value.

  1. Draw a triangle to reflect the given information.
  2. Determine the correct double-angle formula.
  3. Substitute values into the formula based on the triangle.
  4. Simplify.

Example 1

Using a Double-Angle Formula to Find the Exact Value Involving Tangent

Given that tanθ= 3 4 tanθ= 3 4 and θ θ is in quadrant II, find the following:

  1. sin( 2θ ) sin( 2θ )
  2. cos( 2θ ) cos( 2θ )
  3. tan( 2θ ) tan( 2θ )
Try It #1

Given sinα= 5 8 , sinα= 5 8 , with α α in quadrant I, find cos( 2α ). cos( 2α ).

Example 2

Using the Double-Angle Formula for Cosine without Exact Values

Use the double-angle formula for cosine to write cos( 6x ) cos( 6x ) in terms of cos( 3x ). cos( 3x ).

Analysis

This example illustrates that we can use the double-angle formula without having exact values. It emphasizes that the pattern is what we need to remember and that identities are true for all values in the domain of the trigonometric function.

Using Double-Angle Formulas to Verify Identities

Establishing identities using the double-angle formulas is performed using the same steps we used to derive the sum and difference formulas. Choose the more complicated side of the equation and rewrite it until it matches the other side.

Example 3

Using the Double-Angle Formulas to Verify an Identity

Verify the following identity using double-angle formulas:

1+sin( 2θ )= ( sinθ+cosθ ) 2 1+sin( 2θ )= ( sinθ+cosθ ) 2

Analysis

This process is not complicated, as long as we recall the perfect square formula from algebra:

( a±b ) 2 = a 2 ±2ab+ b 2 ( a±b ) 2 = a 2 ±2ab+ b 2

where a=sinθ a=sinθ and b=cosθ. b=cosθ. Part of being successful in mathematics is the ability to recognize patterns. While the terms or symbols may change, the algebra remains consistent.

Try It #2

Verify the identity: cos 4 θ sin 4 θ=cos( 2θ ). cos 4 θ sin 4 θ=cos( 2θ ).

Example 4

Verifying a Double-Angle Identity for Tangent

Verify the identity:

tan( 2θ )= 2 cotθtanθ tan( 2θ )= 2 cotθtanθ

Analysis

Here is a case where the more complicated side of the initial equation appeared on the right, but we chose to work the left side. However, if we had chosen the left side to rewrite, we would have been working backwards to arrive at the equivalency. For example, suppose that we wanted to show

2tanθ 1 tan 2 θ = 2 cotθtanθ 2tanθ 1 tan 2 θ = 2 cotθtanθ

Let’s work on the right side.

2 cotθtanθ = 2 1 tanθ tanθ ( tanθ tanθ ) = 2tanθ 1 tanθ ( tanθ )tanθ(tanθ) = 2tanθ 1 tan 2 θ 2 cotθtanθ = 2 1 tanθ tanθ ( tanθ tanθ ) = 2tanθ 1 tanθ ( tanθ )tanθ(tanθ) = 2tanθ 1 tan 2 θ

When using the identities to simplify a trigonometric expression or solve a trigonometric equation, there are usually several paths to a desired result. There is no set rule as to what side should be manipulated. However, we should begin with the guidelines set forth earlier.

Try It #3

Verify the identity: cos(2θ)cosθ= cos 3 θcosθ sin 2 θ. cos(2θ)cosθ= cos 3 θcosθ sin 2 θ.

Use Reduction Formulas to Simplify an Expression

The double-angle formulas can be used to derive the reduction formulas, which are formulas we can use to reduce the power of a given expression involving even powers of sine or cosine. They allow us to rewrite the even powers of sine or cosine in terms of the first power of cosine. These formulas are especially important in higher-level math courses, calculus in particular. Also called the power-reducing formulas, three identities are included and are easily derived from the double-angle formulas.

We can use two of the three double-angle formulas for cosine to derive the reduction formulas for sine and cosine. Let’s begin with cos( 2θ )=12 sin 2 θ. cos( 2θ )=12 sin 2 θ. Solve for sin 2 θ: sin 2 θ:

cos(2θ) = 12 sin 2 θ 2 sin 2 θ = 1cos(2θ) sin 2 θ = 1cos(2θ) 2 cos(2θ) = 12 sin 2 θ 2 sin 2 θ = 1cos(2θ) sin 2 θ = 1cos(2θ) 2

Next, we use the formula cos( 2θ )=2 cos 2 θ1. cos( 2θ )=2 cos 2 θ1. Solve for cos 2 θ: cos 2 θ:

cos(2θ) =  2 cos 2 θ1 1+cos(2θ) = 2 cos 2 θ 1+cos(2θ) 2 = cos 2 θ cos(2θ) =  2 cos 2 θ1 1+cos(2θ) = 2 cos 2 θ 1+cos(2θ) 2 = cos 2 θ

The last reduction formula is derived by writing tangent in terms of sine and cosine:

tan 2 θ = sin 2 θ cos 2 θ = 1cos(2θ) 2 1+cos(2θ) 2 Substitute the reduction formulas. = ( 1cos(2θ) 2 )( 2 1+cos(2θ) ) = 1cos(2θ) 1+cos(2θ) tan 2 θ = sin 2 θ cos 2 θ = 1cos(2θ) 2 1+cos(2θ) 2 Substitute the reduction formulas. = ( 1cos(2θ) 2 )( 2 1+cos(2θ) ) = 1cos(2θ) 1+cos(2θ)

Reduction Formulas

The reduction formulas are summarized as follows:

sin 2 θ= 1cos( 2θ ) 2 sin 2 θ= 1cos( 2θ ) 2
cos 2 θ= 1+cos( 2θ ) 2 cos 2 θ= 1+cos( 2θ ) 2
tan 2 θ= 1cos( 2θ ) 1+cos( 2θ ) tan 2 θ= 1cos( 2θ ) 1+cos( 2θ )

Example 5

Writing an Equivalent Expression Not Containing Powers Greater Than 1

Write an equivalent expression for cos 4 x cos 4 x that does not involve any powers of sine or cosine greater than 1.

Analysis

The solution is found by using the reduction formula twice, as noted, and the perfect square formula from algebra.

Example 6

Using the Power-Reducing Formulas to Prove an Identity

Use the power-reducing formulas to prove

sin 3 ( 2x )=[ 1 2 sin( 2x ) ][ 1cos( 4x ) ] sin 3 ( 2x )=[ 1 2 sin( 2x ) ][ 1cos( 4x ) ]

Analysis

Note that in this example, we substituted

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

for sin 2 ( 2x ). sin 2 ( 2x ). The formula states

sin 2 θ= 1cos( 2θ ) 2 sin 2 θ= 1cos( 2θ ) 2

We let θ=2x, θ=2x, so 2θ=4x. 2θ=4x.

Try It #4

Use the power-reducing formulas to prove that 10 cos 4 x= 15 4 +5cos( 2x )+ 5 4 cos( 4x ). 10 cos 4 x= 15 4 +5cos( 2x )+ 5 4 cos( 4x ).

Using Half-Angle Formulas to Find Exact Values

The next set of identities is the set of half-angle formulas, which can be derived from the reduction formulas and we can use when we have an angle that is half the size of a special angle. If we replace θ θ with α 2 , α 2 , the half-angle formula for sine is found by simplifying the equation and solving for sin( α 2 ). sin( α 2 ). Note that the half-angle formulas are preceded by a ± ± sign. This does not mean that both the positive and negative expressions are valid. Rather, it depends on the quadrant in which α 2 α 2 terminates.

The half-angle formula for sine is derived as follows:

sin 2 θ = 1cos(2θ) 2 sin 2 ( α 2 ) = 1( cos2 α 2 ) 2 = 1cosα 2 sin( α 2 ) = ± 1cosα 2 sin 2 θ = 1cos(2θ) 2 sin 2 ( α 2 ) = 1( cos2 α 2 ) 2 = 1cosα 2 sin( α 2 ) = ± 1cosα 2

To derive the half-angle formula for cosine, we have

cos 2 θ = 1+cos(2θ) 2 cos 2 ( α 2 ) = 1+cos( 2 α 2 ) 2 = 1+cosα 2 cos( α 2 ) = ± 1+cosα 2 cos 2 θ = 1+cos(2θ) 2 cos 2 ( α 2 ) = 1+cos( 2 α 2 ) 2 = 1+cosα 2 cos( α 2 ) = ± 1+cosα 2

For the tangent identity, we have

tan 2 θ = 1cos(2θ) 1+cos(2θ) tan 2 ( α 2 ) = 1cos(2 α 2 ) 1+cos(2 α 2 ) = 1cosα 1+cosα tan( α 2 ) = ± 1cosα 1+cosα tan 2 θ = 1cos(2θ) 1+cos(2θ) tan 2 ( α 2 ) = 1cos(2 α 2 ) 1+cos(2 α 2 ) = 1cosα 1+cosα tan( α 2 ) = ± 1cosα 1+cosα

Half-Angle Formulas

The half-angle formulas are as follows:

sin( α 2 )=± 1cosα 2 sin( α 2 )=± 1cosα 2
cos( α 2 )=± 1+cosα 2 cos( α 2 )=± 1+cosα 2
tan( α 2 )=± 1cosα 1+cosα = sinα 1+cosα = 1cosα sinα tan( α 2 )=± 1cosα 1+cosα = sinα 1+cosα = 1cosα sinα

Example 7

Using a Half-Angle Formula to Find the Exact Value of a Sine Function

Find sin(15°) sin(15°) using a half-angle formula.

Analysis

Notice that we used only the positive root because sin(15°) sin(15°) is positive.

How To

Given the tangent of an angle and the quadrant in which the angle lies, find the exact values of trigonometric functions of half of the angle.

  1. Draw a triangle to represent the given information.
  2. Determine the correct half-angle formula.
  3. Substitute values into the formula based on the triangle.
  4. Simplify.

Example 8

Finding Exact Values Using Half-Angle Identities

Given that tanα= 8 15 tanα= 8 15 and α α lies in quadrant III, find the exact value of the following:

  1. sin( α 2 ) sin( α 2 )
  2. cos( α 2 ) cos( α 2 )
  3. tan( α 2 ) tan( α 2 )
Try It #5

Given that sinα= 4 5 sinα= 4 5 and α α lies in quadrant IV, find the exact value of cos( α 2 ). cos( α 2 ).

Example 9

Finding the Measurement of a Half Angle

Now, we will return to the problem posed at the beginning of the section. A bicycle ramp is constructed for high-level competition with an angle of θ θ formed by the ramp and the ground. Another ramp is to be constructed half as steep for novice competition. If tanθ= 5 3 tanθ= 5 3 for higher-level competition, what is the measurement of the angle for novice competition?

Media

Access these online resources for additional instruction and practice with double-angle, half-angle, and reduction formulas.

9.3 Section Exercises

Verbal

1.

Explain how to determine the reduction identities from the double-angle identity cos( 2x )= cos 2 x sin 2 x. cos( 2x )= cos 2 x sin 2 x.

2.

Explain how to determine the double-angle formula for tan(2x) tan(2x) using the double-angle formulas for cos(2x) cos(2x) and sin(2x). sin(2x).

3.

We can determine the half-angle formula for tan( x 2 )= 1cosx 1+cosx tan( x 2 )= 1cosx 1+cosx by dividing the formula for sin( x 2 ) sin( x 2 ) by cos( x 2 ). cos( x 2 ). Explain how to determine two formulas for tan( x 2 ) tan( x 2 ) that do not involve any square roots.

4.

For the half-angle formula given in the previous exercise for tan( x 2 ), tan( x 2 ), explain why dividing by 0 is not a concern. (Hint: examine the values of cosx cosx necessary for the denominator to be 0.)

Algebraic

For the following exercises, find the exact values of a) sin( 2x ), sin( 2x ), b) cos( 2x ), cos( 2x ), and c) tan( 2x ) tan( 2x ) without solving for x. x.

5.

If sinx= 1 8 , sinx= 1 8 , and x x is in quadrant I.

6.

If cosx= 2 3 , cosx= 2 3 , and x x is in quadrant I.

7.

If cosx= 1 2 , cosx= 1 2 , and x x is in quadrant III.

8.

If tanx=−8, tanx=−8, and x x is in quadrant IV.

For the following exercises, find the values of the six trigonometric functions if the conditions provided hold.

9.

cos(2θ)= 3 5 cos(2θ)= 3 5 and 90°θ180° 90°θ180°

10.

cos(2θ)= 1 2 cos(2θ)= 1 2 and 180°θ270° 180°θ270°

For the following exercises, simplify to one trigonometric expression.

11.

2sin( π 4 )2cos( π 4 ) 2sin( π 4 )2cos( π 4 )

12.

4sin( π 8 )cos( π 8 ) 4sin( π 8 )cos( π 8 )

For the following exercises, find the exact value using half-angle formulas.

13.

sin( π 8 ) sin( π 8 )

14.

cos( 11π 12 ) cos( 11π 12 )

15.

sin( 11π 12 ) sin( 11π 12 )

16.

cos( 7π 8 ) cos( 7π 8 )

17.

tan( 5π 12 ) tan( 5π 12 )

18.

tan( 3π 12 ) tan( 3π 12 )

19.

tan( 3π 8 ) tan( 3π 8 )

For the following exercises, find the exact values of a) sin( x 2 ), sin( x 2 ), b) cos( x 2 ), cos( x 2 ), and c) tan( x 2 ) tan( x 2 ) without solving for x, x, when x360°. x360°.

20.

If tanx= 4 3 , tanx= 4 3 , and x x is in quadrant IV.

21.

If sinx= 12 13 , sinx= 12 13 , and x x is in quadrant III.

22.

If cscx=7, cscx=7, and x x is in quadrant II.

23.

If secx=4, secx=4, and x x is in quadrant II.

For the following exercises, use Figure 5 to find the requested half and double angles.

Image of a right triangle. The base is length 12, and the height is length 5. The angle between the base and the height is 90 degrees, the angle between the base and the hypotenuse is theta, and the angle between the height and the hypotenuse is alpha degrees.
Figure 5
24.

Find sin( 2θ ),cos(2θ), sin( 2θ ),cos(2θ), and tan(2θ). tan(2θ).

25.

Find sin(2α),cos(2α), sin(2α),cos(2α), and tan(2α). tan(2α).

26.

Find sin( θ 2 ),cos( θ 2 ), sin( θ 2 ),cos( θ 2 ), and tan( θ 2 ). tan( θ 2 ).

27.

Find sin( α 2 ),cos( α 2 ), sin( α 2 ),cos( α 2 ), and tan( α 2 ). tan( α 2 ).

For the following exercises, simplify each expression. Do not evaluate.

28.

cos 2 (28°) sin 2 (28°) cos 2 (28°) sin 2 (28°)

29.

2 cos 2 (37°)1 2 cos 2 (37°)1

30.

12 sin 2 (17°) 12 sin 2 (17°)

31.

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

32.

4sin(8x)cos(8x) 4sin(8x)cos(8x)

33.

6sin(5x)cos(5x) 6sin(5x)cos(5x)

For the following exercises, prove the given identity.

34.

( sintcost ) 2 =1sin( 2t ) ( sintcost ) 2 =1sin( 2t )

35.

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

36.

cotxtanx=2cot( 2x ) cotxtanx=2cot( 2x )

37.

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

For the following exercises, rewrite the expression with an exponent no higher than 1.

38.

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

39.

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

40.

sin 4 (8x) sin 4 (8x)

41.

sin 4 (3x) sin 4 (3x)

42.

cos 2 x sin 4 x cos 2 x sin 4 x

43.

cos 4 x sin 2 x cos 4 x sin 2 x

44.

tan 2 x sin 2 x tan 2 x sin 2 x

Technology

For the following exercises, reduce the equations to powers of one, and then check the answer graphically.

45.

tan 4 x tan 4 x

46.

sin 2 (2x) sin 2 (2x)

47.

sin 2 x cos 2 x sin 2 x cos 2 x

48.

tan 2 xsinx tan 2 xsinx

49.

tan 4 x cos 2 x tan 4 x cos 2 x

50.

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

51.

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

52.

tan 2 ( x 2 )sinx tan 2 ( x 2 )sinx

For the following exercises, algebraically find an equivalent function, only in terms of sinx sinx and/or cosx, cosx, and then check the answer by graphing both functions.

53.

sin(4x) sin(4x)

54.

cos(4x) cos(4x)

Extensions

For the following exercises, prove the identities.

55.

sin( 2x )= 2tanx 1+ tan 2 x sin( 2x )= 2tanx 1+ tan 2 x

56.

cos(2α)= 1 tan 2 α 1+ tan 2 α cos(2α)= 1 tan 2 α 1+ tan 2 α

57.

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

58.

( sin 2 x1 ) 2 =cos( 2x )+ sin 4 x ( sin 2 x1 ) 2 =cos( 2x )+ sin 4 x

59.

sin( 3x )=3sinx cos 2 x sin 3 x sin( 3x )=3sinx cos 2 x sin 3 x

60.

cos( 3x )= cos 3 x3 sin 2 xcosx cos( 3x )= cos 3 x3 sin 2 xcosx

61.

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

62.

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

63.

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

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