Consider the functions $f(x) = A\sin(Bx+C)$ and $g(x) = A\cos(Bx+C)$ .

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

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

When $B = 1$ and $C=0$ and as long as $-1 \le \frac DA \le 1$ this equation has at least one solution $x = \arcsin ( \frac DA)$ or $x = \arccos ( \frac DA)$ and others determined by the facts that for all $x$, $\sin(x+2\pi) =\sin(\pi-x)=\sin(x)$ and $\cos(x+2\pi) =\cos(-x)=\cos(x)$.

When $B\ne 1$ , $B \ne 0$ and $C=0$ the equation can be solved so that $x =\frac {\arcsin(\frac D A)}B$ or $x =\frac {\arccos(\frac D A)}B$.

The next example shows *an important visual connection* between a
mapping diagram for a trigonometric function as a composition
with core linear functions and the algebra used in solving a
trigonometric equation.

You can use this next dynamic example to solve trigonometric equations like those in Examples SEQ.SC.1 through SEQ.SC.3 visually with a mapping diagram of $f$ or $g$ and horizontal lines in the graph of $f$ or $g$.