MATHPOWERTM 12, WESTERN EDITION

5.1

5.1.1

Chapter 5 Trigonometric Equations

General Note about Graphing Calculator Windows

The window variables of a graphing calculator canbe expressed as:

x: [Xmin, Xmax, Xscl] y: [Ymin, Ymax, Yscl]

5.1.2

Therefore, the window is:x: [0, 6.28, 1.57] y: [-4, 2, 1]

For example:

5.1.3

Using a Graph to Solve a Trigonometric Equationy = 3sin 4x

From the graph of y = 3sin 4x, find all the solutions for 3sin 4x = 0, in the interval 0≤ x < 2The solutions to the equation 3sin 4x = 0 are equivalent to finding the x-intercepts of the graph y = 3sin 4x.From the graph, the x-intercepts occur at

0,

4

,2

,34

,,54

,32

, and74

.

Therefore, the solutions for 3sin 4x = 0 are

0,

4

,2

,34

,,54

,32

, and74

.

Using a Graph to Solve a Trigonometric Equation

Find all the solutions to the equation 2cos x - 1 = 0.

Method 1. Solve the equation for cos x and then graph both sides of the equation. The intersection of the two graphs is the solution.cos x

1

2

0 ≤ x < 3600

The solutions for 2cos x - 1 = 0 are600 and 3000.

y = cos xy

1

2

5.1.4

Method 2.

Using a Graph to Solve a Trigonometric Equation [cont’d]

Find all the solutions to the equation 2cos x - 1 = 0. 0 ≤ x < 3600

Graph y = 2cos x - 1. The x-intercepts are the solution.

The solutions for 2cos x - 1 = 0 are 600 and 3000.

5.1.5

Using a Graph to Find Solutions

cos x = 0

x

2

, 32

cos 2x = 0

x 4

, 34

,

54

, 74

The graph of y = cos x has a period of 2. Within the interval 0 ≤ x ≤ 2, there are 2 zeros.

y = cos 2xy = cos x

The graph y = cos 2x has a period of . Therefore, there are two cycles in the interval 0 ≤ x ≤ 2Each cycle has 2 zeros.

5.1.6

cos 3x = 0

x 6

, 36

,

56

, 76

,

96

, 116

Finding the General Solutions

The graph of y = cos x has two x-intercepts in theinterval 0 ≤ x ≤ Therefore, there are two solutions to the equation cos x = 0.

If the restriction of the interval were removed, then the solutions would occur every 2Therefore, the generalsolutions for y = cos x are:

x

2

2 n, n I

5.1.7

x

32

2 n, n I

Finding the General Solutions [cont’d]

2

52

92

132

172

32

72

112

152

x

2

2 n, n I x

32

2 n, n I

5.1.8

x

4

n x

34

n

The graph of y = cos 2x is related to the graph of y = cos x by a horizontal compression factor of 2. Setting 2x to thegeneral solutions for cos x = 0 yields the following solutions:

2x

2

2 n or 2x 32

2 n

x

4

2 n

2 x

34

2 n

2

Finding the General Solutions

x

4

n x

34

n

4

54

94

134

174

214

254

294

334

34

74

114

154

194

234

274

314

354

5.1.9

cos x = 0

x

2

, 32

cos 2x = 0

x 4

, 34

,

54

, 74

Solving Trig Equations

2x 2

+ 2n

2x 32

+ 2n

2x 2

, 32

52

, 72

5.1.10

Suggested Questions:Pages 247 and 2482, 3, 5, 6, 11,12, 16-18,21, 26

5.1.11