We start by recalling the key concept: The inverse of a
function.
We consider the three core trigonometric functions
$sin$, $cos$ and $tan$ and their related inverse functions $\arcsin, \arccos,$ and $ \arctan$.
Functions that have these core trigonometric functions as components
either arithmetically
or as compositions will have inverses using the related inverse
functions. If a core trigonometric function is used, then the related
inverse trigonometric function
will appear in the inverse, and vice versa.
Note: The algebra of trigonometric
and inverse trigonometric functions can sometimes help simplify the
expressions of inverses.
As as example we consider functions of the form $f(x) = A \tan(x-h)+k$ for $x \in (-\frac{\pi}2+h,\frac{\pi}2+h)$.
This trigonometric function is constructed with
compositions using the tangent function, $\tan$ , and core
linear functions. Finding the algebraic expression for the inverse
of $f$ can be interpreted visually by composing the inverses
of $f_{-h}(x)=x-h$, $\tan$ , $f_A(x)
=A*x$,and $f_{+k}(x)=x+k$ in the reverse order ("socks
and shoes").
Thus the inverse of $f(x) = A \tan(x-h)+k$ is
$g(x)= (f_{+h}\circ \arctan \circ f_{1/A}\circ
f_{-k})(x) = \arctan { \frac {x-k} A} +h $.
We use mapping diagrams to verify and visualize how the
function $g(x) = \arctan { \frac {x-k} A} +h $ is the inverse of $f$
when $A \ne 0$ by looking at the composition
functions $(f\circ g)(x) = x$ for $x \in R$ and $(g \circ f)(x) =
x$ for all $x \in (-\frac{\pi}2+h,\frac{\pi}2+h)$.
Example
TRIG.INV.1 :
Suppose $f(x) = 2\tan(x-1) -3$ for $x \in (-\frac{\pi}2+1,\frac{\pi}2+1)$. Verify that $g(x) = \arctan( \frac
{x+3} 2) + 1$ is the inverse function for $f$ .
Inverses for $f(x) =A*trig(x)$
and $f(x)=trig(x) + k$ are quite simple: $g(x) = arctrig( \frac x A )$ and $g(x)= arctrig(x-k)$.
Example
TRIG.INV.2 : (i) Suppose $f(x) =A*trig(x)$ [$A \ne 0$] Verify that $g(x) = arctrig( \frac x A )$ is the inverse function for $f$.
ii) Suppose $f(x)=trig(x) + k$. Verify that $g(x)= arctrig(x-k)$ is the inverse function for $f$.
You can use this next dynamic example
to examine visually $g$ as the inverse of $f$ with the
mapping diagrams and the graphs of $g$ and $f$.
Example
TRIG.DINV.0.1 (Not Yet Done) Dynamic Visualization of The Inverse Function
for General Trigonometric Functions: Graph and Mapping Diagrams