TEF - Inverse Tangent Lesson

Inverse Tangent

Let's review a little bit of information about equation image indicator $LaTeX: f\left(x\right)=\tan x$ $f\left(x\right)=\tan x$ .

1. What is the domain of equation image indicator $LaTeX: f\left(x\right)=\tan x$ $f\left(x\right)=\tan x$ ?

Solution: all real numbers, $LaTeX: x\ne\frac{\pi}{2}+\pi n$ $x\ne\frac{\pi}{2}+\pi n$

2. What is the range of text annotation indicator equation image indicator $LaTeX: f\left(x\right)=\tan x$ $f\left(x\right)=\tan x$ ?

Solution: $LaTeX: \left(-\infty,\:\infty\right)$ $\left(-\infty,\:\infty\right)$

Watch this video to get an idea of how we will restrict the domain of equation image indicator $LaTeX: f\left(x\right)=\tan x$ $f\left(x\right)=\tan x$ to create $LaTeX: f^{\:-1}\left(x\right)=\tan^{-1}x$ $f^{\:-1}\left(x\right)=\tan^{-1}x$ .

So, now that you understand where equation image indicator $LaTeX: f\left(x\right)=\tan^{-1}x$ $f\left(x\right)=\tan^{-1}x$ it is important to know some of these facts:

We can call the inverse function $LaTeX: f\left(x\right)=\tan^{-1}x$ $f\left(x\right)=\tan^{-1}x$ or $LaTeX: f\left(x\right)=arc\:\tan x$ $f\left(x\right)=arc\:\tan x$ .
The domain of $LaTeX: f\left(x\right)=\tan^{-1}x\:is\:\left(-\infty,\:\infty\right)$ $f\left(x\right)=\tan^{-1}x\:is\:\left(-\infty,\:\infty\right)$ .
The range of $LaTeX: f\left(x\right)=\tan^{-1}x\:is\:\left(-\frac{\Pi}{2},\:\frac{\Pi}{2}\right)$ $f\left(x\right)=\tan^{-1}x\:is\:\left(-\frac{\Pi}{2},\:\frac{\Pi}{2}\right)$ .

cycle of inverse tangent
tan(angle)=ratio
tan to -1(ratio)=angle
the input becomes the output
the output becomes the input

tangent unit circle indicating positive tangent and negative tangent

Watch this video to practice a few more problems:

Try these problems to check your understanding:

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