Trigonometric Equations

Basic Trigonometric Identities

• Pythagorean Identity: \(\sin^2(x) + \cos^2(x) = 1\)
• Reciprocal Identities: \(\csc(x) = \frac{1}{\sin(x)}, \sec(x) = \frac{1}{\cos(x)}, \cot(x) = \frac{1}{\tan(x))}\)
• Quotient Identities: \(\tan(x) = \frac{\sin(x)}{\cos(x)}, \cot(x) = \frac{\cos(x)}{\sin(x)}\)
• Even/Odd Identities: \(\sin(-x) = -\sin(x), \cos(-x) = \cos(x), \tan(-x) = -\tan(x)\)

Solving Trigonometric Equations

1. Isolate the trigonometric function on one side of the equation.
2. Use trigonometric identities to simplify the equation.
3. Find the general solution by adding multiples of the period of the function.
4. Check for extraneous solutions by ensuring that the solutions satisfy the original equation.

Example

1. Isolate the sine function: \(x = \sin^{-1}\left(\frac{1}{2}\)\)
2. Using the unit circle or reference angles, we find the solutions \(x = \frac{\pi}{6} + 2n\pi\) and \(x = \frac{5\pi}{6} + 2n\pi\), where \(n\) is an integer.
3. The general solution is \(x = \frac{\pi}{6} + 2n\pi\) and \(x = \frac{5\pi}{6} + 2n\pi\).
4. Check for extraneous solutions by substituting the solutions into the original equation.

Study Guide

• Understand the basic trigonometric identities and how to use them to simplify equations.
• Practice finding solutions to trigonometric equations using reference angles and the unit circle.
• Be familiar with the period and amplitude of trigonometric functions and how they affect the solutions.
• Check for extraneous solutions and ensure that the solutions satisfy the original equation.