5.3 Solving Trigonometric Equations
Solving trigonometric equations is similar to solving algebraic equations in that you may need to:
- isolate the trig function (like isolating the variable)
- combine like terms
- factor and set factors equal to zero
- square both sides of the equation
- use the quadratic formula if necessary
When solving a trig equation, in addition to the typical strategies listed above for algebraic equations, ask
yourself these basic questions…
1. Are the trig functions the same?
2. Are the angles the same or multiples of each other?
3. What is the domain? Is it a given interval or all real numbers?
How you proceed to solve the equation depends on the three critical questions above. In addition to solving trig
equations algebraically, you can solve them on your graphing calculator. Use your graphing calculator to
CHECK the solutions for each example below after solving algebraically.
Day 1:
CASE 1. Are the trig functions the same?
(HINT: Isolate the trig function.)
YES. Only one trig function in the equation.
2 sin x − 3 = 0
0 ≤ x < 2π
(HINT: Use a trig identity to make a
substitution.)
NO. More than one trig function in the equation.
2 sin 2 x + 3 cos x − 3 = 0
0 ≤ x < 2π
sin x + cos x ⋅ cot x = 2
0 ≤ x < 2π
tan 3 x = 3tan x
[0, 2π )
Day 2:
CASE 2. Is the angle “x” or a multiple of “x” (like 2x, 3x, x/2, etc)?
(HINT: If the equation contains a multiple angle 2x, 3x, etc. remember to adjust the domain.)
1.) 2 cos 3t − 1 = 0
3.) tan 2x(tan x − 1)
0 ≤ t < 2π
0 ≤ x < 2π
2.) 4 sin 2 2x = 3
4.) tan
x −1
=
2
3
0 ≤ x < 2π
0 ≤ x < 2π
(HOMEWORK HINT: In homework – do #69 in radians and #73 in degrees)
DAY 3
CASE 3: Solving over all real numbers.
In all of the examples above the domain has been [0,2π ]. Suppose the domain is all real numbers. Write the
solution to your equation using a formula in terms of n for n is an integer.
Examples:
2. 3 tan 2 x − 1 = 0
1. 2 cos x + 3 = 0 Domain : 
Domain : 
Some more complicated trig equations may involve aspects of each of the examples above.
3. 2 sin
x
+1= 0
3
4. sin 2 θ = cos2 θ − sin θ
0 ≤ x < 6π
(HINT: Solve for the trig function and adjust the domain.)
0 ≤ θ < 360
(HINT: Use a trig identity substitution.)
5. 6 sin 2 x + 5 sin x = 6
6. 3cos 2 x − 4 cos x − 2 = 0
domain: all real numbers
Domain : 
(HINT: If equation does not factor, use quadratic formula)