Section 7.3 trigonometric equations

Section 7.3

Trigonometric Equations

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Determine whether is a solution of the equation 2sin 2 0. 4

5Is a solution?4


22sin 2 2 2 2 2 2 2 04 2

is NOT a solution.4

22sin 2 2 2 2 24

052

5 is a solution.4

2cos 3 0


3cos2

3Where is cos ? On the interval 2

0,2 there are two answers, one in

quadrant I and one in quadrant IV.

11On 0, 2 , or =6 6

A general formula would include all coterminal angles112 or = 2

6 6k k

Solve the equation: 2cos 2 1 0, 0 2


1cos 22

1On the interval 0, 2 the cosine of an angle is 2

5when the angle is or .3 3

52 or 2 but since we divide by 2, adding 2 to the angle 3 3

will still give us a solution for that falls in the interval 0,2 .

5 52 , 2 +2 , 2 , 2 23 3 3 3

5 7 11, , , 6 6 6 6

Solve the equation: 3 tan 3 1 0, 0 2


1tan 33

On the interval 0,2 the tangent of an angle is

3 5 11when the angle is or .3 6 6

5 113 or 3 but since we divide by 3, adding 2 and 4 to 6 6

the angle will still give us a solution for that falls in the interval 0,2 .

5 5 5 11 11 113 , 3 +2 , 3 +4 ,3 , 3 2 ,3 46 6 6 6 6 6

5 11 17 23 29 35, , , , , 18 18 18 18 18 18

Solve the equation: cos 1, 0 24


The cosine equals 1 when the angle is 0.

04

4

cos 0.2


1cos 0.2 1.37 For inverse cosine the calculator will only give an angle from 0 to π. There is another angle in the interval from 0 to 2π with that cosine value in quadrant IV.

x

y

1.37 2 1.37

1.37,4.91

2Solve the equation: 2cos cos 1 0, 0 2


2cos 1 cos 1 0

2cos 1 0 or cos 1 0

1cos or cos 12

2 4 , , 03 3

2 40, ,3 3

2 2Solve the equation: sin sin cos , 0 2


2 2sin sin 1 sin 22sin sin 1 0

2sin 1 sin 1 0

2sin 1 0 or sin 1 0

1sin or sin 12

7 11 , ,6 6 2

7 11, ,2 6 6

2 2Solve the equation: 3sin cos , 0 2


2 23sin 1 sin 24sin 1 2 1sin4

1 1sin4 2

6

,56

,76

,11

6

6

,56

, 76

,116

Section 7.3 trigonometric equations

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Transcript of Section 7.3 trigonometric equations