As a general description, there are 3 steps. These steps may be very challenging, or even impossible, depending on the equation.
Step 1: Find the trigonometric values need to be to solve the equation.
Step 2: Find all 'angles' that give us these values from step 1.
Step 3: Find the values of the unknown that will result in angles that we got in step 2.
(Long) Example
Solve: 2sin(4x- pi/3)=1
Step 1: The only trig function in this equation is sin.
Sometimes it is helpful to make things look simpler by replacing, like this:
Replace sin(4x- pi/3) by the single letter S. Now we need to find S to make 2S=1. Simple! Make S=1/2
So a solution will need to make sin(4x- pi/3)=1/2
Step 2: The 'angle' in this equation is (4x- pi/3). For the moment, let's call that theta. We need sin theta = 1/2
There are infinitely many such theta, we need to find them all.
Every theta that makes sin theta = 1/2 is coterminal with either pi/6 or with (5 pi)/6. (Go through one period of the graph, or once around the unit circle.)
So theta Which, remember is our short way of writing 4x- pi/3 must be of the form: theta = pi/6+2 pi k for some integer k or of the form theta = (5 pi)/6 +2 pi k for some integer k.
Step 3:
Replacing theta in the last bit of step 2, we see that we need one of: 4x- pi/3 = pi/6+2 pi k for integer k
or 4x- pi/3 = (5 pi)/6+2 pi k for integer k.
Adding pi/3 in the form (2 pi)/6 to both sides of these equations gives us:
4x = (3 pi)/6+2 pi k = pi/2+2 pi k for integer k or
4x = (7 pi)/6+2 pi k for integer k.
Dividing by 4 (multiplying by 1/4) gets us to:
x= pi/8+(2pi k)/4 or
x=(7 pi)/24+(2 pi k)/4 for integer k.
We can write this in simpler form:
x= pi/8+pi/2 k or
x=(7 pi)/24+pi/2 k for integer k.
Final note The Integer k could be a positive or negative whole number or 0. If k is negative, we're actually subtracting from the basic solution.