Solving Trigonometric Equations

T, 11.0: Students demonstrate an understanding of half-angle and double-angle formulas for sines and cosines and can use those formulas to prove and/or simplify other trigonometric identities.

Solving Trigonometric Equations

Objectives• Solve trigonometric

equations on a restricted domain and justify each step

• Use the periodicity of the trigonometric functions to find the general solutions of a trigonometric equation.

Key Words• Quadrant• Principle Values

– Solutions are restricted to two adjacent quadrants

Pre-requisite Check

Solve for x

– We have . So x=0 or x=.

1. 3x=2– Then

– We have . So x=1 or x=.

Solve for angle

– Using the unit circle or the properties of the cosine graph:

Example 1

Solve cos x + 2 tan x cos2 x = 0 for principal values of x. Express solutions in degrees.

3 cos x + 2 tan x cos2 x = 0

3 cos x + 2(sin xcos x) cos2 x = 0 tan x =

sin xcos x

3 cos x + 2 sin x cos x = 0 cos x ( 3 + 2 sin x) = 0 Factor. cos x = 0 x = 90°

3 + 2 sin x = 0

sin x = -3

2

x = -60°

Example 2

Solve

tan x sin2 x + tan x cos2 x = tan2 x

for 0 ≤ ≤ 2.

This equation can be written in terms of tan x only. tan x sin2 x + tan x cos2 x = tan2 x tan x (sin2 x + cos2 x) = tan2 x Factor. tan x (1) = tan2 x Pythagorean identity: sin2 x + cos2 x = 1 tan x - tan2 x = 0 tan x ( 1 - tan x) = 0 Factor. tan x = 0 x = 0 or x =

1 - tan x = 0 tan x = 1

x = 4 or x =

54

The solutions are 0, 4, , and

54 .

Example 3

Solve

cos2 x sin x - sin x + cos2 x - = 0

for all real values of x.

Use factoring to solve this equation.

cos2 x sin x - 12 sin x + cos2 x -

12 = 0

sin x ( cos2 x - 12) + cos2 x -

12 = 0 Factor.

( cos2 x - 12)(sin x + 1) = 0

cos2 x - 12 = 0

cos2 x = 12

cos x = 2

2

x = 4 +

2k

sin x + 1 = 0 sin x = -1

x = 32 + 2k

The solutions are 4 +

2k and

32 + 2k.

Example 4

Solve

2 cos - < 0

for 0 ≤ ≤ 2.

In terms of the unit circle, we need to find

points with x-coordinates less than 3

2 .

The values of for which cos < 3

2 are

6 and

116 . The figure shows that the

solution of the inequality is 6 ≤ ≤

116 .

Conclusions

Summary• Solve cos2x – cosx + 1 = 0

for 0≤x<360.– 60,90,270,300

Assignment• Solving Trigonometric

Equations– Page 459– #(5-16)