# Solving Trig Equations

# 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.

## Final Advice

- 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.