5.5 Solving Trigonometric Equations
Example 1
A) Is
4
x
3
B) Is
x

3
a solution to
1
cos x  
2
?
a solution to cos x = sin 2x ?
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
2
sin x  
2
Example 2 - Solution
5
7
x
 2 n or x 
 2 n
4
4
where n is any integer
Example 3
Solve the equation on the interval [0º , 360º)
sin x = 1 2
x = 30º , 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
Ans : x 
 2n and
 2n
6
6
Example 5
Solve the equation on the interval [0 , 2π)
2 cos x − 1 = 0
2 cos x = 1
cos x = 1
2
x=
 5
3
,
3
Example 6 -Trigonometric Equations
Quadratic in Form.
Solve the equation: 2sin2 x  3sin x  1 0; 0  x  2
Try to solve by factoring
It factors in the same manner as 2 x 2  3x  1
= (2x -1)(x – 1)
Solution: (2sinx – 1)(sinx -1) = 0
2sinx – 1 = 0
2sinx = 1
Therefore x = π/6, 5π/6
sinx – 1 = 0
sinx = 1
Ans. π/6, π/2, 5π/6
sinx = ½
x = π/2
Example 7: Solve : 2sin 2   1  0 over 0    2
2sin   1
2
sin 2   1/ 2
1
sin   
2
2
sin   
2

 3 5 7
4
,
4
,
4
,
4
Example 8: Solve an Equation with a
Multiple Angle.
Solve the equation: tan2 x  3
0  x  2

2 7 5
Ans : ,
,
,
6 3 6
3
Example 9 - Multiple Angle
x 1
Solve the equation: sin  ; 0  x  2
3 2
Ans. x =

2
Example 10
Solve the equation: tan x sin x  3 tan x; 0  x  2
2
Move all terms to one side, then factor out a common trig
function.
Ans. 0, π
Example 11
Solve the equation: 2sin2 x  3cos x  0
0  x  2
The equation contains more than one
trig function; there is no common trig
function. Try using an identity.
Ans. π/3, 5π/3
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
Use a calculator to find the reference angle, then
use your knowledge of signs of trigonometric
functions to find x in the required interval.
Ans. a. 1.2592, 4.4008
b. 3.3752, 6.0496
Example 15
Solve the equation: cos2 x  5cos x  3  0; 0  x  2
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
Ans. 2.3423, 3.9409