5.3 – Solving Trigonometric Equations

In this section, you will learn to:Use standard algebraic techniques to

solve trigonometric equations

Solve trigonometric equations of quadratic type

Solve trigonometric equations involving multiple angles

Use inverse trigonometric functions to solve trigonometric equations

Two basic techniques for solving trigonometric equations are factoring and applying known identities.

a) Rewrite the equation with a single trigonometric function.

b) Remember only to factor after you have set the equation equal to zero.

Collect Like Terms/Take the Square Root:

2tan 1 3x 1) Solve 2 tan 1 3 and find all solutions.x

2 tan 2x

3 7and between 0,2

4 4x

tan 1x

3

4x n

Collect Like Terms/Take the Square Root:

(Solve for extraneous roots.)

2sec 2 0x 22) Solve sec 2 0 in the interval 0,2 .x

sec 2x

2sec 2x

3 5 7, , ,

4 4 4 4x

2 3 5 7cos , , ,

2 4 4 4 4x x

Use identities to solve:

2 2sin cosx x

2 23) Solve sin cos for all solutions.x x

2 2sin cos 0x x

2 2sin 1 sin 0x x

2sin

2x

22sin 1 0x

2 1sin

2x

Set the equation to zero tofactor.

Substitute in known identities.

Simplify and solve.

4 2x n

Find all solutions for .x

Use identities to solve: 24) Solve 2sin cos 1 0 for the interval 0,2 .x x

22sin cos 1 0 Substitute in known identities.x x

22 2cos cos 1 0 Simplify and combine like terms.x x

22 1 cos cos 1 0 Simplify and combine like terms.x x

22cos cos 1 0 Multiply by 1.x x 22cos cos 1 0 Factor and solve.x x

2cos 1 cos 1 0x x 1

cos and cos 1 Find all solutions for .2

x x x

5 5, , ,

3 3 3 3x x x

Factor to solve: 4 25) Solve sin 3sin 4 0 for the interval 0,2 .x x

Factor to solve:4 2sin 3sin 4 0 Factorx x

2 2sin 4 sin 1 0 Solve separatelyx x

2 2sin 4 sin 1x x

no solution since sine

oscillates between 1 and 1

x

sin 2 sin 1x x

Solving of Multiple Angles:

6) Solve 2sin 4 3 0 for all solutions.x

2sin 4 3 0x

3sin 4

2x

24 4

3 3x and x

12 6x and x

2 , 212 6

x n n

HomeworkPage 376-37935-47 odd, 49-53 odd, 61-71 odd