Trigonometric Equations and General Solutions

A trigonometric equation is an equation involving trigonometric ratios of an unknown angle, such as $\sin\theta=\tfrac12$. Because every trigonometric ratio is periodic, such an equation is satisfied not by a single value of $\theta$ but by infinitely many values, spaced one full period apart. The value found by restricting $\theta$ to its smallest possible range (commonly $0$ to $2\pi$, or $-\pi/2$ to $\pi/2$ for $\sin$/$\tan$) is called the principal solution; expressing ALL solutions in one formula, valid for every integer $n$, is called the general solution.

The three standard general solutions (with $n \in \mathbb{Z}$, the set of integers) are:

$\tan\theta = \tan\alpha \implies \theta = n\pi + \alpha$ …