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)=12sin(4x−π3)=1
Step 1: The only trig function in this equation is sinsin.
Sometimes it is helpful to make things look simpler by replacing, like this:
Replace sin(4x- pi/3)sin(4x−π3) by the single letter SS. Now we need to find SS to make 2S=12S=1. Simple! Make S=1/2S=12
So a solution will need to make sin(4x- pi/3)=1/2sin(4x−π3)=12
Step 2: The 'angle' in this equation is (4x- pi/3)(4x−π3). For the moment, let's call that thetaθ. We need sin theta = 1/2sinθ=12
There are infinitely many such thetaθ, we need to find them all.
Every thetaθ that makes sin theta = 1/2sinθ=12 is coterminal with either pi/6π6 or with (5 pi)/65π6. (Go through one period of the graph, or once around the unit circle.)
So thetaθ Which, remember is our short way of writing 4x- pi/34x−π3 must be of the form: theta = pi/6+2 pi kθ=π6+2πk for some integer kk or of the form theta = (5 pi)/6 +2 pi kθ=5π6+2πk for some integer kk.
Step 3:
Replacing thetaθ in the last bit of step 2, we see that we need one of: 4x- pi/3 = pi/6+2 pi k4x−π3=π6+2πk for integer kk
or 4x- pi/3 = (5 pi)/6+2 pi k4x−π3=5π6+2πk for integer kk.
Adding pi/3π3 in the form (2 pi)/62π6 to both sides of these equations gives us:
4x = (3 pi)/6+2 pi k = pi/2+2 pi k4x=3π6+2πk=π2+2πk for integer kk or
4x = (7 pi)/6+2 pi k4x=7π6+2πk for integer kk.
Dividing by 44 (multiplying by 1/414) gets us to:
x= pi/8+(2pi k)/4x=π8+2πk4 or
x=(7 pi)/24+(2 pi k)/4x=7π24+2πk4 for integer kk.
We can write this in simpler form:
x= pi/8+pi/2 kx=π8+π2k or
x=(7 pi)/24+pi/2 kx=7π24+π2k for integer kk.
Final note The Integer kk could be a positive or negative whole number or 0. If kk is negative, we're actually subtracting from the basic solution.