Solving Trig Equations

Basic principles

Trigonometric equations are equations involving one or multiple trigonometric functions.
The solutions of a trigonometric equation are the values of the angles that make the equation hold true.

Step 1: Isolate the trigonometric term. This means making the term with sin, cos or tan the subject of the equation.
Step 2: Determine the reference angle. This is the angle you find in the range 0 to 90 degrees that solves the equation. Typically, you calculate this by using a calculator to apply the inverse trig function.
Step 3: Determine all the solutions within 0 to 360 degrees or 0 to 2π. Use knowledge of the trig function’s behaviour - for example, knowing that sin(θ) = sin(180 - θ) could help you find a second solution.

Equations including cos, sin, and tan within the same equation can often be reduced to simpler equations through the application of Pythagorean identity or double-angle identities.
If an equation includes multiple instances of the same trig function, you can often turn it into a quadratic equation - these are familiar from earlier work, and can be solved in familiar ways.
Be sure to check all potential solutions in the original equation. Occasionally, methods for solving multiple-term equations can produce extraneous solutions which do not solve the original equation.

CAST is a mnemonic that helps remember the signs of the trig functions in each quadrant – in the 1st quadrant (0-90 degrees), all are positive, in the 2nd quadrant (90-180 degrees) only Sin is positive, in the 3rd quadrant (180-270 degrees) only Tan is positive, and in the 4th quadrant (270-360 degrees) only Cos is positive.
Combining knowledge of CAST with the unit circle allows for the calculation of sine, cosine and tangent of angles greater than 90 degrees or less than 0 degrees.

Always use exact values where possible. These include well-known values of sin, cos, and tan at angles like 30, 45, 60, and 90 degrees.
Keep in mind the periodic nature of trig functions - i.e., the fact that their graphs repeat every 360 degrees or 2π radians. This means there are an infinite number of solutions to most trig equations.
To specify solutions within a certain range, you first find the solutions in range 0 to 360 degrees or 0 to 2π, and then add to these the solutions in each subsequent or previous period as needed.