5.5 Solving Trigonometric EquationsExample 1

A) Is a solution to ?

B) Is a solution to cos x = sin 2x ?

4

3x

1cos

2x

3x

Solving Trigonometric Equations - Overview

Trigonometric Equations with a Single Trig Function

• For equations with a single trig function, isolate the trig function on one side.

• Solve for the variable by identifying the appropriate angles.

• Be prepared to express your answer in radian measure.

Example 2

Find all solutions for 2sin

2x

Example 2 - Solution

where n is any integer

5 72 2

4 4x n or x n

Example 3

Solve the equation on the interval [0º , 360º) sin x =

x = 30º

21

, 150º

Other Strategies for Solving

Solving Trig Equations

• Put the equation in terms of one trig function (if possible).

• Solve for the trig function (using algebra – addition, subtraction, multiplication, division, factoring).

• Solve for the variable (using inverse trig functions, reference angles).

• Use a fundamental identity to end up with a single trig function.

Example 4

To solve an equation containing a single trig function:

Solve: 3sinx – 2 = 5sinx - 1* Isolate the function on one side of the equation. * Solve for the variable.

Solution: 3sinx - 5sinx = -1 +2 -2sinx = 1 sinx = -1/2 (Remember: x are the angles whose sine is -1/2) 7 11

: 2 and 26 6

Ans x n n

Example 5

Solve the equation on the interval [0 , 2π)

2 cos x − 1 = 0

2 cos x = 1 cos x =

x =

1

2

5,

3 3

Example 6 -Trigonometric Equations Quadratic in Form.

2Solve the equation: 2sin 3sin 1 0; 0 2x x x

Ans. π/6, π/2, 5π/6

Try to solve by factoringIt factors in the same manner as= (2x -1)(x – 1)

Solution: (2sinx – 1)(sinx -1) = 0 2sinx – 1 = 0 2sinx = 1 sinx = ½Therefore x = π/6, 5π/6

sinx – 1 = 0 sinx = 1 x = π/2

22 3 1x x

2Example7: : 2sin 1 0 0 2Solve over

22sin 1 2sin 1/ 2

1sin

2

2sin

2

3 5 7, , ,

4 4 4 4

Example 8: Solve an Equation with a Multiple Angle.

Solve the equation: tan2 3 0 2x x

2 7 5: , , ,

6 3 6 3Ans

1Solve the equation: sin ; 0 2

3 2

xx

Ans. x =

Example 9 - Multiple Angle

2

2Solve the equation: tan sin 3 tan ; 0 2x x x x

Ans. 0, π

Move all terms to one side, then factor out a common trig function.

Example 10

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

Ans. π/3, 5π/3

The equation contains more than one trig function; there is no common trig function. Try using an identity.

Example 11

Example 12

Solve the equation: cos2x + 3sinx – 2 = 0, 0 ≤ x ≤ 2π

Ans. π/6, π/2, and 5π/6

Example 13

Solve the equation: sinx cosx= -1/2, 0 ≤ x ≤ 2π

Ans. 3π/4, 7π/4

Example 14 - using a calculator to solve

Solve the equation correct to four decimal places, 0 ≤ x ≤ 2π a. tan x = 3.1044 b. sin x = -0.2315

Ans. a. 1.2592, 4.4008 b. 3.3752, 6.0496

Use a calculator to find the reference angle, then use your knowledge of signs of trigonometric functions to find x in the required interval.

2Solve the equation: cos 5cos 3 0; 0 2x x x

Ans. 2.3423, 3.9409

The equation is in quadratic form, but does not factor. Use the quadratic formula to solve for the trig function of x, then use a calculator and the

Example 15