TRIG.SEQ.T Solving     Tangent Equations
 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..
Example  SEQ.T.1 : Suppose $10 \tan(x)= 5$. Find $x$.
Example SEQ.T.2 :Suppose $10\tan(4x)= 5$. Find $x$.
Now consider 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.T.3 : Suppose $4* \tan(2x-1) +1 = 9$. Find $x$.
This 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 $\tan$.
You can use this next dynamic example to solve trigonometric equations like those in Examples SEQ.T.1 and SEQ.T.2 visually with a mapping diagram of $f$ and the lines in the graph of $f$.
Example TRIG.DSEQ.T.0 Dynamic Views for solving an equation $f(x) = A\tan(Bx) = D$ on Graphs and Mapping Diagrams