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Algebra and Trigonometry

9.2 Sum and Difference Identities

Algebra and Trigonometry9.2 Sum and Difference Identities
  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 sum and difference formulas for cosine.
  • Use sum and difference formulas for sine.
  • Use sum and difference formulas for tangent.
  • Use sum and difference formulas for cofunctions.
  • Use sum and difference formulas to verify identities.
Photo of Mt. McKinley.
Figure 1 Mount McKinley, in Denali National Park, Alaska, rises 20,237 feet (6,168 m) above sea level. It is the highest peak in North America. (credit: Daniel A. Leifheit, Flickr)

How can the height of a mountain be measured? What about the distance from Earth to the sun? Like many seemingly impossible problems, we rely on mathematical formulas to find the answers. The trigonometric identities, commonly used in mathematical proofs, have had real-world applications for centuries, including their use in calculating long distances.

The trigonometric identities we will examine in this section can be traced to a Persian astronomer who lived around 950 AD, but the ancient Greeks discovered these same formulas much earlier and stated them in terms of chords. These are special equations or postulates, true for all values input to the equations, and with innumerable applications.

In this section, we will learn techniques that will enable us to solve problems such as the ones presented above. The formulas that follow will simplify many trigonometric expressions and equations. Keep in mind that, throughout this section, the term formula is used synonymously with the word identity.

Using the Sum and Difference Formulas for Cosine

Finding the exact value of the sine, cosine, or tangent of an angle is often easier if we can rewrite the given angle in terms of two angles that have known trigonometric values. We can use the special angles, which we can review in the unit circle shown in Figure 2.

Diagram of the unit circle with points labeled on its edge. P point is at an angle a from the positive x axis with coordinates (cosa, sina). Point Q is at an angle of B from the positive x axis with coordinates (cosb, sinb). Angle POQ is a - B degrees. Point A is at an angle of (a-B) from the x axis with coordinates (cos(a-B), sin(a-B)). Point B is just at point (1,0). Angle AOB is also a - B degrees. Radii PO, AO, QO, and BO are all 1 unit long and are the legs of triangles POQ and AOB. Triangle POQ is a rotation of triangle AOB, so the distance from P to Q is the same as the distance from A to B.
Figure 2 The Unit Circle

We will begin with the sum and difference formulas for cosine, so that we can find the cosine of a given angle if we can break it up into the sum or difference of two of the special angles. See Table 1.

Sum formula for cosine cos( α+β )=cosαcosβsinαsinβ cos( α+β )=cosαcosβsinαsinβ
Difference formula for cosine cos( αβ )=cosαcosβ+sinαsinβ cos( αβ )=cosαcosβ+sinαsinβ
Table 1

First, we will prove the difference formula for cosines. Let’s consider two points on the unit circle. See Figure 3. Point P P is at an angle α α from the positive x-axis with coordinates ( cosα,sinα ) ( cosα,sinα ) and point Q Q is at an angle of β β from the positive x-axis with coordinates ( cosβ,sinβ ). ( cosβ,sinβ ). Note the measure of angle POQ POQ is αβ. αβ.

Label two more points: A A at an angle of ( αβ ) ( αβ ) from the positive x-axis with coordinates ( cos( αβ ),sin( αβ ) ); ( cos( αβ ),sin( αβ ) ); and point B B with coordinates ( 1,0 ). ( 1,0 ). Triangle POQ POQ is a rotation of triangle AOB AOB and thus the distance from P P to Q Q is the same as the distance from A A to B. B.

Diagram of the unit circle with points labeled on its edge. P point is at an angle a from the positive x axis with coordinates (cosa, sina). Point Q is at an angle of B from the positive x axis with coordinates (cosb, sinb). Angle POQ is a - B degrees. Point A is at an angle of (a-B) from the x axis with coordinates (cos(a-B), sin(a-B)). Point B is just at point (1,0). Angle AOB is also a - B degrees. Radii PO, AO, QO, and BO are all 1 unit long and are the legs of triangles POQ and AOB. Triangle POQ is a rotation of triangle AOB, so the distance from P to Q is the same as the distance from A to B.
Figure 3

We can find the distance from P P to Q Q using the distance formula.

d PQ = (cosαcosβ) 2 + (sinαsinβ) 2 = cos 2 α2cosαcosβ+ cos 2 β+ sin 2 α2sinαsinβ+ sin 2 β d PQ = (cosαcosβ) 2 + (sinαsinβ) 2 = cos 2 α2cosαcosβ+ cos 2 β+ sin 2 α2sinαsinβ+ sin 2 β

Then we apply the Pythagorean identity and simplify.

= ( cos 2 α+ sin 2 α )+( cos 2 β+ sin 2 β )2cosαcosβ2sinαsinβ = 1+12cosαcosβ2sinαsinβ = 22cosαcosβ2sinαsinβ = ( cos 2 α+ sin 2 α )+( cos 2 β+ sin 2 β )2cosαcosβ2sinαsinβ = 1+12cosαcosβ2sinαsinβ = 22cosαcosβ2sinαsinβ

Similarly, using the distance formula we can find the distance from A A to B. B.

d AB = (cos(αβ)1) 2 + (sin(αβ)0) 2 = cos 2 (αβ)2cos(αβ)+1+ sin 2 (αβ) d AB = (cos(αβ)1) 2 + (sin(αβ)0) 2 = cos 2 (αβ)2cos(αβ)+1+ sin 2 (αβ)

Applying the Pythagorean identity and simplifying we get:

= ( cos 2 (αβ)+ sin 2 (αβ) )2cos(αβ)+1 = 12cos(αβ)+1 = 22cos(αβ) = ( cos 2 (αβ)+ sin 2 (αβ) )2cos(αβ)+1 = 12cos(αβ)+1 = 22cos(αβ)

Because the two distances are the same, we set them equal to each other and simplify.

22cosαcosβ2sinαsinβ = 22cos(αβ) 22cosαcosβ2sinαsinβ = 22cos(αβ) 22cosαcosβ2sinαsinβ = 22cos(αβ) 22cosαcosβ2sinαsinβ = 22cos(αβ)

Finally we subtract 2 2 from both sides and divide both sides by −2. −2.

cosαcosβ+sinαsinβ=cos(αβ)   cosαcosβ+sinαsinβ=cos(αβ)  

Thus, we have the difference formula for cosine. We can use similar methods to derive the cosine of the sum of two angles.

Sum and Difference Formulas for Cosine

These formulas can be used to calculate the cosine of sums and differences of angles.

cos(α+β)=cosαcosβsinαsinβ cos(α+β)=cosαcosβsinαsinβ
cos(αβ)=cosαcosβ+sinαsinβ cos(αβ)=cosαcosβ+sinαsinβ

How To

Given two angles, find the cosine of the difference between the angles.

  1. Write the difference formula for cosine.
  2. Substitute the values of the given angles into the formula.
  3. Simplify.

Example 1

Finding the Exact Value Using the Formula for the Cosine of the Difference of Two Angles

Using the formula for the cosine of the difference of two angles, find the exact value of cos( 5π 4 π 6 ). cos( 5π 4 π 6 ).

Try It #1

Find the exact value of cos( π 3 π 4 ). cos( π 3 π 4 ).

Example 2

Finding the Exact Value Using the Formula for the Sum of Two Angles for Cosine

Find the exact value of cos(75°). cos(75°).

Analysis

Note that we could have also solved this problem using the fact that 75°=135°60°. 75°=135°60°.

cos(αβ) = cosαcosβ+sinαsinβ cos(135°60°) = cos(135°)cos(60°)+sin(135°)sin(60°) = ( 2 2 )( 1 2 )+( 2 2 )( 3 2 ) = ( 2 4 )+( 6 4 ) = ( 6 2 4 ) cos(αβ) = cosαcosβ+sinαsinβ cos(135°60°) = cos(135°)cos(60°)+sin(135°)sin(60°) = ( 2 2 )( 1 2 )+( 2 2 )( 3 2 ) = ( 2 4 )+( 6 4 ) = ( 6 2 4 )
Try It #2

Find the exact value of cos(105°). cos(105°).

Using the Sum and Difference Formulas for Sine

The sum and difference formulas for sine can be derived in the same manner as those for cosine, and they resemble the cosine formulas.

Sum and Difference Formulas for Sine

These formulas can be used to calculate the sines of sums and differences of angles.

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

How To

Given two angles, find the sine of the difference between the angles.

  1. Write the difference formula for sine.
  2. Substitute the given angles into the formula.
  3. Simplify.

Example 3

Using Sum and Difference Identities to Evaluate the Difference of Angles

Use the sum and difference identities to evaluate the difference of the angles and show that part a equals part b.

  1. sin(45°30°) sin(45°30°)
  2. sin(135°120°) sin(135°120°)

Example 4

Finding the Exact Value of an Expression Involving an Inverse Trigonometric Function

Find the exact value of sin( cos −1 1 2 + sin −1 3 5 ). sin( cos −1 1 2 + sin −1 3 5 ). Then check the answer with a graphing calculator.

Using the Sum and Difference Formulas for Tangent

Finding exact values for the tangent of the sum or difference of two angles is a little more complicated, but again, it is a matter of recognizing the pattern.

Finding the sum of two angles formula for tangent involves taking quotient of the sum formulas for sine and cosine and simplifying. Recall, tanx= sinx cosx ,cosx0. tanx= sinx cosx ,cosx0.

Let’s derive the sum formula for tangent.

tan(α+β) = sin(α+β) cos(α+β) = sinαcosβ+cosαsinβ cosαcosβsinαsinβ = sinαcosβ+cosαsinβ cosαcosβ cosαcosβsinαsinβ cosαcosβ Divide the numerator and denominator by cosαcosβ. = sinα cosβ cosα cosβ + cosα sinβ cosα cosβ cosα cosβ cosα cosβ sinαsinβ cosαcosβ = sinα cosα + sinβ cosβ 1 sinαsinβ cosαcosβ = tanα+tanβ 1tanαtanβ tan(α+β) = sin(α+β) cos(α+β) = sinαcosβ+cosαsinβ cosαcosβsinαsinβ = sinαcosβ+cosαsinβ cosαcosβ cosαcosβsinαsinβ cosαcosβ Divide the numerator and denominator by cosαcosβ. = sinα cosβ cosα cosβ + cosα sinβ cosα cosβ cosα cosβ cosα cosβ sinαsinβ cosαcosβ = sinα cosα + sinβ cosβ 1 sinαsinβ cosαcosβ = tanα+tanβ 1tanαtanβ

We can derive the difference formula for tangent in a similar way.

Sum and Difference Formulas for Tangent

The sum and difference formulas for tangent are:

tan( α+β )= tanα+tanβ 1tanαtanβ tan( α+β )= tanα+tanβ 1tanαtanβ
tan( αβ )= tanαtanβ 1+tanαtanβ tan( αβ )= tanαtanβ 1+tanαtanβ

How To

Given two angles, find the tangent of the sum of the angles.

  1. Write the sum formula for tangent.
  2. Substitute the given angles into the formula.
  3. Simplify.

Example 5

Finding the Exact Value of an Expression Involving Tangent

Find the exact value of tan( π 6 + π 4 ). tan( π 6 + π 4 ).

Try It #3

Find the exact value of tan( 2π 3 + π 4 ). tan( 2π 3 + π 4 ).

Example 6

Finding Multiple Sums and Differences of Angles

Given  sinα= 3 5 ,0<α< π 2 ,cosβ= 5 13 ,π<β< 3π 2 ,  sinα= 3 5 ,0<α< π 2 ,cosβ= 5 13 ,π<β< 3π 2 , find

  1. sin( α+β ) sin( α+β )
  2. cos( α+β ) cos( α+β )
  3. tan( α+β ) tan( α+β )
  4. tan( αβ ) tan( αβ )

Analysis

A common mistake when addressing problems such as this one is that we may be tempted to think that α α and β β are angles in the same triangle, which of course, they are not. Also note that

tan( α+β )= sin( α+β ) cos( α+β ) tan( α+β )= sin( α+β ) cos( α+β )

Using Sum and Difference Formulas for Cofunctions

Now that we can find the sine, cosine, and tangent functions for the sums and differences of angles, we can use them to do the same for their cofunctions. You may recall from Right Triangle Trigonometry that, if the sum of two positive angles is π 2 , π 2 , those two angles are complements, and the sum of the two acute angles in a right triangle is π 2 , π 2 , so they are also complements. In Figure 6, notice that if one of the acute angles is labeled as θ, θ, then the other acute angle must be labeled ( π 2 θ ). ( π 2 θ ).

Notice also that sinθ=cos( π 2 θ ), sinθ=cos( π 2 θ ), which is opposite over hypotenuse. Thus, when two angles are complementary, we can say that the sine of θ θ equals the cofunction of the complement of θ. θ. Similarly, tangent and cotangent are cofunctions, and secant and cosecant are cofunctions.

Image of a right triangle. The remaining angles are labeled theta and pi/2 - theta.
Figure 6

From these relationships, the cofunction identities are formed. Recall that you first encountered these identities in The Unit Circle: Sine and Cosine Functions.

Cofunction Identities

The cofunction identities are summarized in Table 2.

sinθ=cos( π 2 θ ) sinθ=cos( π 2 θ ) cosθ=sin( π 2 θ ) cosθ=sin( π 2 θ )
tanθ=cot( π 2 θ ) tanθ=cot( π 2 θ ) cotθ=tan( π 2 θ ) cotθ=tan( π 2 θ )
secθ=csc( π 2 θ ) secθ=csc( π 2 θ ) cscθ=sec( π 2 θ ) cscθ=sec( π 2 θ )
Table 2

Notice that the formulas in the table may also justified algebraically using the sum and difference formulas. For example, using

cos( αβ )=cosαcosβ+sinαsinβ, cos( αβ )=cosαcosβ+sinαsinβ,

we can write

cos( π 2 θ ) = cos π 2 cosθ+sin π 2 sinθ = (0)cosθ+(1)sinθ = sinθ cos( π 2 θ ) = cos π 2 cosθ+sin π 2 sinθ = (0)cosθ+(1)sinθ = sinθ

Example 7

Finding a Cofunction with the Same Value as the Given Expression

Write tan π 9 tan π 9 in terms of its cofunction.

Try It #4

Write sin π 7 sin π 7 in terms of its cofunction.

Using the Sum and Difference Formulas to Verify Identities

Verifying an identity means demonstrating that the equation holds for all values of the variable. It helps to be very familiar with the identities or to have a list of them accessible while working the problems. Reviewing the general rules presented earlier may help simplify the process of verifying an identity.

How To

Given an identity, verify using sum and difference formulas.

  1. Begin with the expression on the side of the equal sign that appears most complex. Rewrite that expression until it matches the other side of the equal sign. Occasionally, we might have to alter both sides, but working on only one side is the most efficient.
  2. Look for opportunities to use the sum and difference formulas.
  3. Rewrite sums or differences of quotients as single quotients.
  4. If the process becomes cumbersome, rewrite the expression in terms of sines and cosines.

Example 8

Verifying an Identity Involving Sine

Verify the identity sin(α+β)+sin(αβ)=2sinαcosβ. sin(α+β)+sin(αβ)=2sinαcosβ.

Example 9

Verifying an Identity Involving Tangent

Verify the following identity.

sin(αβ) cosαcosβ =tanαtanβ sin(αβ) cosαcosβ =tanαtanβ
Try It #5

Verify the identity: tan( πθ )=tanθ. tan( πθ )=tanθ.

Example 10

Using Sum and Difference Formulas to Solve an Application Problem

Let L 1 L 1 and L 2 L 2 denote two non-vertical intersecting lines, and let θ θ denote the acute angle between L 1 L 1 and L 2 . L 2 . See Figure 7. Show that

tanθ= m 2 m 1 1+ m 1 m 2 tanθ= m 2 m 1 1+ m 1 m 2

where m 1 m 1 and m 2 m 2 are the slopes of L 1 L 1 and L 2 L 2 respectively. (Hint: Use the fact that tan θ 1 = m 1 tan θ 1 = m 1 and tan θ 2 = m 2 . tan θ 2 = m 2 . )

Diagram of two non-vertical intersecting lines L1 and L2 also intersecting the x-axis. The acute angle formed by the intersection of L1 and L2 is theta. The acute angle formed by L2 and the x-axis is theta 1, and the acute angle formed by the x-axis and L1 is theta 2.
Figure 7

Example 11

Investigating a Guy-wire Problem

For a climbing wall, a guy-wire R R is attached 47 feet high on a vertical pole. Added support is provided by another guy-wire S S attached 40 feet above ground on the same pole. If the wires are attached to the ground 50 feet from the pole, find the angle α α between the wires. See Figure 8.

Two right triangles. Both share the same base, 50 feet. The first has a height of 40 ft and hypotenuse S. The second has height 47 ft and hypotenuse R. The height sides of the triangles are overlapping. There is a B degree angle between R and the base, and an a degree angle between the two hypotenuses within the B degree angle.
Figure 8

Analysis

Occasionally, when an application appears that includes a right triangle, we may think that solving is a matter of applying the Pythagorean Theorem. That may be partially true, but it depends on what the problem is asking and what information is given.

Media

Access these online resources for additional instruction and practice with sum and difference identities.

9.2 Section Exercises

Verbal

1.

Explain the basis for the cofunction identities and when they apply.

2.

Is there only one way to evaluate cos( 5π 4 )? cos( 5π 4 )? Explain how to set up the solution in two different ways, and then compute to make sure they give the same answer.

3.

Explain to someone who has forgotten the even-odd properties of sinusoidal functions how the addition and subtraction formulas can determine this characteristic for f(x)=sin(x) f(x)=sin(x) and g(x)=cos(x). g(x)=cos(x). (Hint: 0x=x 0x=x )

Algebraic

For the following exercises, find the exact value.

4.

cos( 7π 12 ) cos( 7π 12 )

5.

cos( π 12 ) cos( π 12 )

6.

sin( 5π 12 ) sin( 5π 12 )

7.

sin( 11π 12 ) sin( 11π 12 )

8.

tan( π 12 ) tan( π 12 )

9.

tan( 19π 12 ) tan( 19π 12 )

For the following exercises, rewrite in terms of sinx sinx and cosx. cosx.

10.

sin( x+ 11π 6 ) sin( x+ 11π 6 )

11.

sin( x 3π 4 ) sin( x 3π 4 )

12.

cos( x 5π 6 ) cos( x 5π 6 )

13.

cos( x+ 2π 3 ) cos( x+ 2π 3 )

For the following exercises, simplify the given expression.

14.

csc( π 2 t ) csc( π 2 t )

15.

sec( π 2 θ ) sec( π 2 θ )

16.

cot( π 2 x ) cot( π 2 x )

17.

tan( π 2 x ) tan( π 2 x )

18.

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

19.

tan( 3 2 x )tan( 7 5 x ) 1+tan( 3 2 x )tan( 7 5 x ) tan( 3 2 x )tan( 7 5 x ) 1+tan( 3 2 x )tan( 7 5 x )

For the following exercises, find the requested information.

20.

Given that sina= 2 3 sina= 2 3 and cosb= 1 4 , cosb= 1 4 , with a a and b b both in the interval [ π 2 ,π ), [ π 2 ,π ), find sin(a+b) sin(a+b) and cos(ab). cos(ab).

21.

Given that sina= 4 5 , sina= 4 5 , and cosb= 1 3 , cosb= 1 3 , with a a and b b both in the interval [ 0, π 2 ), [ 0, π 2 ), find sin(ab) sin(ab) and cos(a+b). cos(a+b).

For the following exercises, find the exact value of each expression.

22.

sin( cos 1 (0) cos 1 ( 1 2 ) ) sin( cos 1 (0) cos 1 ( 1 2 ) )

23.

cos( cos 1 ( 2 2 )+ sin 1 ( 3 2 ) ) cos( cos 1 ( 2 2 )+ sin 1 ( 3 2 ) )

24.

tan( sin 1 ( 1 2 ) cos 1 ( 1 2 ) ) tan( sin 1 ( 1 2 ) cos 1 ( 1 2 ) )

Graphical

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical. Confirm your answer using a graphing calculator.

25.

cos( π 2 x ) cos( π 2 x )

26.

sin(πx) sin(πx)

27.

tan( π 3 +x ) tan( π 3 +x )

28.

sin( π 3 +x ) sin( π 3 +x )

29.

tan( π 4 x ) tan( π 4 x )

30.

cos( 7π 6 +x ) cos( 7π 6 +x )

31.

sin( π 4 +x ) sin( π 4 +x )

32.

cos( 5π 4 +x ) cos( 5π 4 +x )

For the following exercises, use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think 2x=x+x. 2x=x+x. )

33.

f( x )=sin( 4x )sin( 3x )cosx,g( x )=sinxcos( 3x ) f( x )=sin( 4x )sin( 3x )cosx,g( x )=sinxcos( 3x )

34.

f( x )=cos( 4x )+sinxsin( 3x ),g( x )=cosxcos( 3x ) f( x )=cos( 4x )+sinxsin( 3x ),g( x )=cosxcos( 3x )

35.

f( x )=sin( 3x )cos( 6x ),g( x )=sin( 3x )cos( 6x ) f( x )=sin( 3x )cos( 6x ),g( x )=sin( 3x )cos( 6x )

36.

f(x)=sin(4x),g(x)=sin(5x)cosxcos(5x)sinx f(x)=sin(4x),g(x)=sin(5x)cosxcos(5x)sinx

37.

f(x)=sin(2x),g(x)=2sinxcosx f(x)=sin(2x),g(x)=2sinxcosx

38.

f( θ )=cos( 2θ ),g( θ )= cos 2 θ sin 2 θ f( θ )=cos( 2θ ),g( θ )= cos 2 θ sin 2 θ

39.

f(θ)=tan(2θ),g(θ)= tanθ 1+ tan 2 θ f(θ)=tan(2θ),g(θ)= tanθ 1+ tan 2 θ

40.

f(x)=sin(3x)sinx,g(x)= sin 2 (2x) cos 2 x cos 2 (2x) sin 2 x f(x)=sin(3x)sinx,g(x)= sin 2 (2x) cos 2 x cos 2 (2x) sin 2 x

41.

f(x)=tan(x),g(x)= tanxtan(2x) 1tanxtan(2x) f(x)=tan(x),g(x)= tanxtan(2x) 1tanxtan(2x)

Technology

For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.

42.

sin(75°) sin(75°)

43.

sin(195°) sin(195°)

44.

cos(165°) cos(165°)

45.

cos(345°) cos(345°)

46.

tan(−15°) tan(−15°)

Extensions

For the following exercises, prove the identities provided.

47.

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

48.

tan(a+b) tan(ab) = sinacosa+sinbcosb sinacosasinbcosb tan(a+b) tan(ab) = sinacosa+sinbcosb sinacosasinbcosb

49.

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

50.

cos( x+y )cos( xy )= cos 2 x sin 2 y cos( x+y )cos( xy )= cos 2 x sin 2 y

51.

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

For the following exercises, prove or disprove the statements.

52.

tan(u+v)= tanu+tanv 1tanutanv tan(u+v)= tanu+tanv 1tanutanv

53.

tan(uv)= tanutanv 1+tanutanv tan(uv)= tanutanv 1+tanutanv

54.

tan( x+y ) 1+tanxtanx = tanx+tany 1 tan 2 x tan 2 y tan( x+y ) 1+tanxtanx = tanx+tany 1 tan 2 x tan 2 y

55.

If α, β, α, β, and γ γ are angles in the same triangle, then prove or disprove sin( α+β )=sinγ. sin( α+β )=sinγ.

56.

If α,β, α,β, and y y are angles in the same triangle, then prove or disprove tanα+tanβ+tanγ=tanαtanβtanγ tanα+tanβ+tanγ=tanαtanβtanγ

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