7.4: Basic Trigonometric Equations

Equations like \[\sin^2\theta + \cos^2 \theta = 1\] are identities since it's true for any value of $\theta \in \mathbb{R}$.

But equations like \[2\sin \theta - 1 = 0 \qquad\qquad \tan 2\theta -1 = 0\] are not identities. They are true for specific values of $\theta$.

Goal: Solve simple trigonometric equations like the ones above.

Basic Trigonometric Equations

The general strategy for solving trigonometric equations is two steps:

Solving Trigonometric equations:
  1. Find the solutions in one period.
  2. Find all solutions (accounting for period).
Solve $\sin \theta = \dfrac{1}{2}$.
Solve $\cos \theta = -\dfrac{\sqrt{2}}{2}$.
Solve $\tan \theta = -1$.
Solve $2 \sin \theta - 1 = 0$.
Solve $\tan^2\theta - 3 = 0$.

Solving Trigonometric Equations by Factoring

You may need to simplify before solving.

Solve \[2\cos^2 \theta - 7\cos \theta + 3 = 0\]
Solve \[4\sin\theta \cos\theta + 2 \cos\theta = 0\]