Mathematics 101 Transitional Mathematics

Study Guide :: Unit 8

Trigonometric Identities and Equations

Trigonometric identities and equations are both equations having at least one of the six trigonometric functions in them with the angle being the unknown quantity. The difference between an identity equation and an equation which is not an identity is the following:

1. The solution set of an identity equation is the complete set of all possible values of the unknown. In the case of trigonometric identities, the equation will be valid for all angles, whether expressed in degrees or in radians. For example, ${\displaystyle {\sin }^2\theta +{\cos }^2\theta =1}$ is a trigonometric identity equation because it is valid for all angles ${\displaystyle \theta }$.
2. The solution set of an equation which is not an identity is a subset of all possible values of the unknown. In the case of trigonometric equations, this means that there will be angles which do not satisfy the equation. For example, ${\displaystyle {\sin }^2\theta =1}$ is a trigonometric equation which is not an identity because there is at least one angle which does not satisfy the equation, namely ${\displaystyle \theta =0}$ radians—but there are many more.

We will study identities first because, in some cases, a trigonometric identity can help solve a trigonometric equation.

Learning Objectives

After you have completed Unit 8, you should be able to:

1. define a trigonometric identity and a trigonometric equation and distinguish between them;
2. recognize and apply in context what are generally termed the Reciprocal, the Cofunction, the Addition and Subtraction, the Double-Angle, and the Half-Angle Formulas of trigonometric functions;
3. prove certain trigonometric identities and solve certain trigonometric equations for both one period and general solutions; and
4. model and solve real-world situations using trigonometric identities and equations.

Readings and Practice

Chapter 8 (pp. 353–393) of the textbook

Optional Sections

• Focus on Modeling: Traveling and Standing Waves (pp. 393–397)

Online graphing tool: Graph

Trigonometric Identities

Textbook Readings

pp. 354–358

Practice

(pp. 358–360):  1, 2, 3, 7, 11, 15, 21, 23, 25, 29–89 odd numbers (choose 8 of the 31 to prove algebraically; note that by graphing each side of an identity, you may confirm its validity by checking if both graphs are coincident), 91, 97, 102 (algebraically prove one of the cofunction identities on page 354)

Answers

pp. 525–526

Terms to Understand

identity equation; trigonometric identity; trigonometric equation

Figures/Tables

Fundamental Trigonometric Identities (p. 354)
Guidelines for Proving Trigonometric Identities (p. 355)

Online Tutorials—the Overview

Pertinent to “Trigonometric Identities”

• Trigonometric Identities
  Trigonometric Identities 1
  Trigonometric Identities 2
  Trigonometric Identities 3

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

Addition and Subtraction Formulas

Optional Subsections

• Evaluating Expressions Involving Inverse Trigonometric Functions (pp. 362–364)
• Expressions of the Form ${\displaystyle A\sin x+B\cos x}$ (pp. 364–365)

For those preparing for Calculus, it is important to have some familiarity with the techniques associated with the inverse trigonometric functions.

Textbook Readings

pp. 360–362

Practice

(pp. 365–367):  1, 2, 3, 7, 11, 13, 15, 17, 23, 27, 31, 37, 53, 69

Answers

pp. 526–527

Terms to Understand

addition and subtraction formulas for sine, cosine, and tangent functions

Figures/Tables

Addition and Subtraction Formulas (p. 360)

The formulas for the sine and cosine sum/difference of angles are important to know.

Online Tutorials—the Overview

Pertinent to “Addition and Subtraction Formulas”

• None

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the ‘Trigonometric Equations’ topic page.

Double-Angle and Half-Angle Formulas

Optional Subsections

• Evaluating Expressions Involving Inverse Trigonometric Functions (pp. 372)
• Product-Sum Formulas (pp. 372–374)

Textbook Readings

pp. 367–372

Practice

Practice (pp. 374–377):  1, 2, 3, 5, 7, 9, 13, 19, 25, 27, 29, 37, 41, 51, 73, 105

Answers

p. 527

Terms to Understand

double-angle and half-angle formulas for sine, cosine, and tangent functions

Figures/Tables

Double-Angle Formulas (p. 368)
Formulas for Lowering Powers (p. 370)
Half-Angle Formulas (p. 370)

The double-angle formulas for the sine and cosine are important to know.

Note that the half-angle formulas can be derived from the double-angle formulas.

Online Tutorials—the Overview

Pertinent to “Double-Angle and Half-Angle Formulas”

• None

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the ‘Trigonometric Equations’ topic page.

Basic Trigonometric Equations

Textbook Readings

pp. 377–382

Practice

(pp. 382–383):  1, 2, 5, 7, 13, 17, 27, 31, 37, 41, 53, 59

Answers

p. 528

Terms to Understand

one period solution set of a trigonometric equation; general solution set (that is, all solutions) of a trigonometric equation

Online Tutorials—the Overview

Pertinent to “Basic Trigonometric Equations”

• Solving Trigonometric Equations I

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the ‘Trigonometric Equations’ topic page.

More Trigonometric Equations

Textbook Readings

pp. 384–388

Practice

(pp. 388–389):  1, 2, 3, 9, 15, 17, 21, 25, 33, 41, 43, 47

Answers

p. 528

Online Tutorials—the Overview

Pertinent to “More Trigonometric Equations”

• Solving Trigonometric Equations II

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the ‘Trigonometric Equations’ topic page.

Unit 8 Review

(pp. 390–391):  odd numbers

Answers

pp. 528–529

Unit 8 Test

(p. 392):  1–4, 6–8 only

Answers

p. 529

Figures/Tables

Formulas to Remember (pp. iv and v at the beginning of the text and pp. 569–570 at the end of the text)