Consider the function $f(x) = A\tan(Bx+C)$.

A standard form of a trigonometric equation with variable $x$ is an equation of the form $f(x) = A\tan(Bx+C) = D$ with $A, B \ne 0$.

Trigonometry courses spend a considerable amount of time solving this type of equation.

When $B = 1$ and $C=0$ this equation has at least one solution $x = \arctan ( \frac DA)$ and others determined by the fact that $\tan(x+\pi) =\tan(x)$.

When $B\ne 1$ , $B \ne 0$ and $C=0$ the equation can be solved so that the key solution is $x =\frac {\arctan(\frac D A)}B$

The following examples illustrate these solutions with both mapping diagrams and graphs..

This example shows

The included GeoGebra mapping diagram can be used to visualize the algebra for solving equations of the form $f(x) = A\cdot trig(Bx-C) +k = D$ with $A,B \ne 0$ and $trig = \sin, \cos$ or $\tan$.