Solving Trigonometric Equations
Big Idea: A trigonometric equation is an equation that involves trigonometric functions such as $\sin\theta$, $\cos\theta$, or $\tan\theta$. Solving these equations means finding all angles $\theta$ that make the equation true.
📐 Key Definitions
Trigonometric Equation: An equation containing one or more trigonometric functions of a variable angle.
Principal Solution: The solution(s) that lie within one full period of the trigonometric function (usually $[0°, 360°)$ or $[0, 2\pi)$).
General Solution: The complete set of all solutions, expressed using a parameter $n$ (where $n$ is any integer) to account for the periodic nature of trig functions.
🔁 Periodicity Reminders
| Function | Period | General Solution Pattern |
|---|---|---|
| $\sin\theta = k$ | $360°$ | $\theta = \alpha + 360°n$ or $\theta = (180° - \alpha) + 360°n$ |
| $\cos\theta = k$ | $360°$ | $\theta = \pm\alpha + 360°n$ |
| $\tan\theta = k$ | $180°$ | $\theta = \alpha + 180°n$ |
Here $\alpha$ is the reference angle and $n \in \mathbb{Z}$.