TRIG.SEQ.SC Sine and Cosine Equations  
 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 $x =\frac {\arccos(\frac D A)}B$.
Example  SEQ.SC.1 : Suppose $10 \sin(x)= 5$. Find $x$.
Example SEQ.SC.2 :Suppose $10\sin(4x)= 5$. Find $x$.
Example SEQ.SC.3 :Suppose $10\cos(4x)= 5$. Find $x$.

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.

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 $\tan$.

Example SEQ.SC.4 Suppose $4\cdot \sin(2x-1) +1 = 3$. Find $x$.

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 $g$.
Example TRIG.DSEQ.SC.0 Dynamic Views for solving an equation $f(x) = A\sin(Bx) = D$ or $f(x) = A\cos(Bx) = D$ on Graphs and Mapping Diagrams