# 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.

## Steps to solve basic trig equations

• 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.

## Solving equations involving multiple trig terms

• 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.

## Using CAST and the Unit Circle

• 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.