MATHPOWERTM 12, WESTERN EDITION

5.2

5.2.1

Chapter 5 Trigonometric Equations

Solving Trigonometric Equations

Quadrant IQuadrant II

Quadrant III Quadrant IV

Cosine

AllSine

Tangent

CAST Rule

Find the measure of 0 ≤ < 3600

a) cos = -0.6691

The reference angle is 480. The angle is foundin Quadrants II and III.

1320 and 2280

b) tan = 1.2435The reference angle is 510. The angle is foundin Quadrants I and III.

510 and 2310

5.2.2

1

2

1

2

3

2

1

3

3

23

1

21

1

2

Exact Values of Special Angles

5.2.3

State the exact value of each ratio:

a) sin56

b) cos34

c) sec76

d) sin32 e) csc f) tan

2

g) csc83

h) cot2

i) sec 3

Finding Exact Values

1

2

2

2

2 3

3

1

2 3

30 1

5.2.4

= undefined = undefined

Evaluate.

a) tan2 116

b) sec2 23

3

3

2

1

3

2 2

4

Finding Exact Values

5.2.5

Related Angles

300

450

600

Quadrants

I II III IV

6

4

3

1500

1350

1200

2100

2250

2400

3300

3150

3000

56

34

23

76

54

43

116

74

53

5.2.6

5.2.7

Solving Trigonometric Equations

sin 3

2

3

3

,23

a) cos 3

2

6

56

,76

b) tan 1

4

4

,54

c)

Solve for if 0 ≤ < 2.

General Solutions

3

2n, 23

2n

n I

ReferenceAngle

ReferenceAngle

ReferenceAngle

General Solutions

56

2n, 76

2n

n I

General Solutions

4

2n, 54

2n

n I

5.2.8

sec 2

3

3

,53

sec 2

4

34

,54

e) cot 3

3

3

23

,53

f)d)

Solving Trigonometric Equations

Solve for if 0 ≤ < 2.

ReferenceAngle

ReferenceAngle

ReferenceAngle

General Solutions

3

2n,

=53

2n

n I

General Solutions

34

2n,

=54

2n

n I

General Solutions

23

2n,

=53

2n

n I

sin2 1

2

4

4

,34

,54

,74

g)

sin 1

2

sin 2

2

Solving Trigonometric Equations

csc2 4

3

3

3

,23

,43

,53

csc 4

3

csc 2

3

h)

5.2.9

Solve for if 0 ≤ < 2.

ReferenceAngle

ReferenceAngle

sin 1

2

Solving Trigonometric Equations

Solve for if 0 ≤ < 2.

a) 3 csc 2 0

3 csc 2

csc 2

3

3

, 23

b) 4 cos 3 2 cos 22 cos 1

cos 1

2

23

, 43

5.2.10

c) d)

e)

NO solution forcos = 3. 5.2.11

ReferenceAngle

ReferenceAngle

Solving Trigonometric Equations

sin 12

6

0

76

116

, , ,

sin 12

or sin 1

6

656

32

, ,

cos 12

or

353

,

2 02sin sin sin ( sin ) 2 1 0 sin sin 0 2 1 0or 2 1 0 1 0sin sin or

2 1 02sin sin ( sin )(sin )2 1 1 0

( cos )(cos )2 1 3 0 2 7 3 02cos cos

2 1 0 3 0cos cos or

cos 3

f)

Reference Angle:

Therefore:

g)

The equation cannotbe factored. Therefore,use the quadratic equation to find the roots:

Reference Angles:

5.2.12

Solving Trigonometric Equations

sin 13

tan b b ac

a

2 42

tan( ) ( ) ( )( )

( )

1 1 4 3 12 3

2

4 1sin sin 3 1sin

0 3398.

0 3398 2 8018. .and

3 1 02tan tan

tan . tan . 0 43 0 76or

0 4061. 0 6499.

2 7355 5 877 0 6499 3 7915. , . , . , .

5.2.13

3 tan2 tan 1 0

Using a Graphing Calculator to Solve Trigonometric Equations

Therefore, and

Suggested Questions:Pages 252 and 2531, 4, 6, 10, 15, 22,24, 27, 32,33 b, 34 a (graph)35 b

5.2.14