Page | 89

Chapter 5 – Analytic Trigonometry

Section 1 Using Fundamental Identities

Section 2 Verifying Trigonometric Identities

Section 3 Solving Trigonometric Equations

Section 4 Sum and Difference Formulas

Section 5 Multiple-Angle and Product-to-Sum Formulas

Vocabulary

Identity Sum and difference formulas

Reduction formulas Multiple-angle formulas

Page |90

What you should learn:

How to recognize and write

the fundamental trigonometric

identities

Section 5.1 Using Fundamental Identities

Objective: In this lesson you learned how to use fundamental trigonometric identities to evaluate

trigonometric functions and simplify trigonometric expressions.

I. Introduction

Name four ways in which the fundamental trigonometric

identities can be used:

1)

2)

3)

4)

List the Fundamental Trigonometric Identities List the six reciprocal identities List the six cofunction identities

1) 1)

2) 2)

3) 3)

4) 4)

5) 5)

6) 6)

Page | 91

What you should learn:

How to use the fundamental

trigonometric identities to

evaluate trigonometric

functions, simplify

trigonometric expressions, and

rewrite trigonometric

expressions

List the two quotient identities List the six even/odd identities

1) 1)

2) 2)

List the three Pythagorean identities 3)

1)

4)

2)

5)

3)

6)

II. Using the Fundamental Identities

Example 1: Explain how to use the fundamental trigonometric

identities to find the value of tan 𝑢 given that sec𝑢 = 2.

Example 2: Explain how to use the fundamental trigonometric identities to simplify sec 𝑥 −

tan 𝑥 sin𝑥.

Page |92

Section 5.1 Examples – Using Fundamental Identities

( 1 ) Use the given values to evaluate (if possible) all six trigonometric functions.

sin 𝑥 =1

2 cos 𝑥 =

√3

2

( 2 ) Use the fundamental identities to simplify the expression.

sin𝜃 (csc𝜃 − sin𝜃)

( 3 ) Factor the expression and use the fundamental identities to simplify.

cot2 𝑥 − cot2 𝑥 cos2 𝑥

( 4 ) Perform the multiplication and use the fundamental identities to simplify.

(sin 𝑥 + cos 𝑥)2

( 5 ) Use trigonometric substitution to write the algebraic expression as a trigonometric function of 𝜃, where

0 < 𝜃 <𝜋

2.

√25 − 𝑥2, 𝑥 = 5 sin 𝜃

Page | 93

What you should learn:

How to understand the

difference between

conditional equations and

identities

What you should learn:

How to verify trigonometric

identities

Section 5.2 Verifying Trigonometric Identities

Objective: In this lesson you learned how to verify trigonometric identities

I. Introduction

The key to both verifying identities and solving equations is:

An identity is:

II. Verifying Trigonometric Identities

Complete the following list of guidelines for verifying

trigonometric identities:

1)

2)

3)

4)

5)

Important Vocabulary

Identity

Page |94

III. Exponent Properties Review

Complete the following:

𝑎𝑚 ∙ 𝑎𝑛 = _________ (𝑎𝑚)𝑛 = __________

𝑎𝑚

𝑎𝑛 = ____________ 𝑎−𝑛 = ____________

𝑎0 = ___________

Page | 95

Section 5.2 Examples – Verifying Trigonometric Identities

( 1 ) Verify the identity.

a) sin 𝑡 csc 𝑡 = 1

b) sin1 2⁄ 𝑥 cos 𝑥 − sin5 2⁄ 𝑥 cos𝑥 = cos3 𝑥 √sin 𝑥

c) cos𝜃

1−sin𝜃= sec 𝜃 + tan𝜃

d) 2 sec2 𝑥 − 2 sec2 𝑥 sin2 𝑥 − sin2 𝑥 − cos2 𝑥 = 1

Page |96

What you should learn:

How to use standard algebraic

techniques to solve

trigonometric equations

What you should learn:

How to solve trigonometric

equations of quadratic type

What you should learn:

How to solve trigonometric

equations involving multiple

angles

Section 5.3 Solving Trigonometric Equations

Objective: In this lesson you learned how to use standard algebraic techniques and inverse

trigonometric functions to solve trigonometric equations.

I. Introduction

To solve a trigonometric equation:

The preliminary goal in solving trigonometric equations is:

How many solutions does the equation sec 𝑥 = 2 have? Explain.

To solve an equation in which two or more trigonometric functions occur:

II. Equations of a Quadratic Type

Give an example of a trigonometric equation of a quadratic

type.

To solve a trigonometric equation of quadratic type:

Care must be taken when squaring each side of a trigonometric equation to obtain a quadratic

because:

III. Functions Involving Multiple Angles

Give an example of a trigonometric function of multiple

angles.

Page | 97

Section 5.3 Examples – Solving Trigonometric Equations

( 1 ) Verify that each 𝑥-value is a solution of the equation.

2 cos 𝑥 − 1 = 0

a) 𝑥 =𝜋

3 b) 𝑥 =

5𝜋

3

( 2 ) Find all solutions of the equation in the intervals [0°, 360°) and [0, 2𝜋).

sin 𝑥 = −√2

2

( 3 ) Solve the equation.

3 sec2 𝑥 − 4 = 0

( 4 ) Find all solutions of the equation in the interval [0, 2𝜋).

cos3 𝑥 = cos 𝑥

Page |98

What you should learn:

How to use sum and difference

formulas to evaluate

trigonometric functions, to

verify identities and to solve

trigonometric equations

Important Vocabulary

Sum and Difference Formulas Reduction Formulas

Section 5.4 Sum and Difference Formulas

Objective: In this lesson you learned how to use sum and difference formulas to rewrite and

evaluate trigonometric functions.

I. Using Sum and Difference Formulas

List the sum and difference formulas for sine, cosine, and

tangent.

sin(𝑢 + 𝑣) = ___________________________________

sin(𝑢 − 𝑣) = ___________________________________

cos(𝑢 + 𝑣) = ___________________________________

cos(𝑢 − 𝑣) = ___________________________________

tan(𝑢 + 𝑣) = ___________________________________

tan(𝑢 − 𝑣) = ___________________________________

A reduction formula is:

Page | 99

Section 5.4 Examples – Sum and Difference Formulas

( 1 ) Find the exact value of each expression.

a) cos(240° − 0°) b) cos 240° − cos0°

( 2 ) Find the exact values of the sine, cosine, and tangent of the angle.

165° = 135° + 30°

( 3 ) Write the expression as the sine, cosine, or tangent of an angle.

cos 60° cos 10° − sin60° sin10°

( 4 ) Find the exact value of the expression without using a calculator.

sin [𝜋

2+ sin−1(−1)]

Page |100

What you should learn:

How to use multiple-angle

formulas to rewrite and

evaluate trigonometric

functions

What you should learn:

How to use power-reducing

formulas to rewrite and

evaluate trigonometric

functions

Section 5.5 Multiple-Angle and Product-to-Sum Formulas

Objective: In this lesson you learned how to use multiple-angle formulas, power-reducing formulas,

half-angle formulas, and product-to-sum formulas to rewrite and evaluate trigonometric

functions.

I. Multiple-Angle Formulas

The most commonly used multiple-angle formulas are the

__________________________, which are listed below:

sin2𝑢 = __________________________

cos 2𝑢 = __________________________

= __________________________

= __________________________

tan 2𝑢 = __________________________

To obtain other multiple-angle formulas:

II. Power-Reducing Formulas

The double-angle formulas can be used to obtain the

__________________________.

The power-reducing formulas are:

sin2 𝑢 = ___________________

cos2 𝑢 = ___________________

tan2 𝑢 = ___________________

Important Vocabulary

Multiple-Angle Formulas

Page | 101

What you should learn:

How to use half-angle formulas

to rewrite and evaluate

trigonometric functions

What you should learn:

How to use product-to-sum

and sum-to-product formulas

to rewrite and evaluate

trigonometric functions

III. Half-Angle Formulas

List the half-angle formulas:

sin𝑢

2= ___________________

cos𝑢

2= ___________________

tan𝑢

2= ___________________= ___________________

The signs of sin𝑢

2 and cos

𝑢

2 depend on:

IV. Product-to-Sum Formulas

The product-to-sum formulas are used in calculus to:

The product-to-sum formulas are:

sin𝑢 sin 𝑣 = _________________________

cos 𝑢 cos 𝑣 = _________________________

sin𝑢 cos 𝑣 = _________________________

cos 𝑢 sin𝑣 = _________________________

The sum-to-product formulas can be used to:

The sum-to-product formulas are:

sin𝑢 + sin 𝑣 = _________________________

sin𝑢 − sin 𝑣 = _________________________

cos 𝑢 + cos 𝑣 = _________________________

cos 𝑢 − cos 𝑣 = _________________________

Page |102

Section 5.5 Examples – Multiple-Angle and Product-to-Sum Formulas

( 1 ) Find the exact values of sin2𝑢, cos2𝑢, and tan 2𝑢 using the double-angle formulas.

sin𝑢 =3

5, 0 < 𝑢 <

𝜋

2

( 2 ) Use a double-angle formula to rewrite the expression.

8 sin𝑥 cos 𝑥

( 3 ) Find the exact values of sin𝑢

2, cos

𝑢

2, and tan

𝑢

2 using the half-angle formulas.

cos 𝑢 =3

5, 0 < 𝑢 <

𝜋

2

( 4 ) Find all solutions of the equation in the interval [0, 2𝜋).

sin 6𝑥 + sin2𝑥 = 0