Trigonometric Equation

Problem 204: Solve the equation $\sin \theta = \tan \theta$ .

Solution: When solving trigonometric equation, we use the same method as solving algebraic equations i.e., solving for $x$ . Thus

(1) $\begin{align*} \sin \theta & = \tan \theta \\ \sin \theta & = \frac{\sin \theta}{\cos \theta}\\ \cos \theta \cdot \sin \theta & = \sin \theta \\ \sin \theta \cdot \cos \that - \sin \theta & = 0 \\ \sin \theta \left(\cos \theta - 1 \right) & = 0 \\ \sin \theta & = 0 \ \text{or} \ \cos \theta = 1 \\ \theta & = 0, \pi \ \text{or} \ \theta = 0 .\end{align*}$

That is, $\theta = 0, \pi$ . However, these are not the only solutions since any other multiple of $2\pi$ will also be a solution.

Hence,

$\theta = 0 + 2k \pi \quad \text{and} \quad \theta = \pi + 2k \pi \quad \text{or} \quad \theta_k = \pi k, \ k = 0, \pm 1, \pm 2, \dots.\quad$

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