Inverse Trig Functions

Inverse Trig Functions

A LevelAQAEdexcelOCRAQA 2022Edexcel 2022OCR 2022

Inverse Trig Functions

We’ve mentioned a little bit about the inverse trig functions already, but now it’s time to take a look at how their graphs look.

We have:

  • \sin ^{-1} known as \arcsin
  • \cos ^{-1} known as \arccos
  • \tan ^{-1} known as \arctan
A Level AQA Edexcel OCR

Setting up the Inversion

Just before we begin, we need to remember that the \textcolor{blue}{\sin}, \textcolor{limegreen}{\cos} and \textcolor{red}{\tan} graphs are not one-to-one functions, they are many-to-one functions. We have a set of x values which give back the same result.

For example, we have \textcolor{blue}{\sin} 45 = \textcolor{blue}{\sin} 135 = \textcolor{blue}{\sin} 405 = \textcolor{blue}{\sin} 495 = ...

A function can only have an inverse if it is one-to-one. As a result, we’ll need to restrict the domain (the range of values of x) of each original trig graph to be one-to-one:

  • \textcolor{blue}{\sin x} is restricted to
    • \dfrac{-\pi}{2} \leq x \leq \dfrac{\pi}{2}
    • -1 \leq \textcolor{blue}{\sin x} \leq 1
  • \textcolor{limegreen}{\cos x} is restricted to
    • 0 \leq x \leq \pi
    • -1 \leq \textcolor{limegreen}{\cos x} \leq 1
  • \textcolor{red}{\tan x} is restricted to
    • \dfrac{-\pi}{2} < x < \dfrac{\pi}{2}
    • The range of \textcolor{red}{\tan x} is unrestricted

 

Here’s the \textcolor{blue}{\sin x} graph, along with \textcolor{purple}{\sin ^{-1}x}

… and now the \textcolor{limegreen}{\cos x} graph, along with \textcolor{purple}{\cos ^{-1}x}

… and the \textcolor{red}{\tan x} graph, along with \textcolor{purple}{\tan ^{-1}x}.

A LevelAQAEdexcelOCR

Note:

Inverse trig functions are NOT the same as the reciprocal trig functions.

So,

  • \sin ^{-1}x \neq \dfrac{1}{\textcolor{blue}{\sin x}}

 

  • \cos ^{-1}x \neq \dfrac{1}{\textcolor{limegreen}{\cos x}}

 

  • \tan ^{-1}x \neq \dfrac{1}{\textcolor{red}{\tan x}}

Inverse Representation in Graphical Form

Actually, you might have noticed that the inverse function is the original function reflected in the line y = x.

This is true for any inverse function, but this is probably a good time to mention it anyway.

A LevelAQAEdexcelOCR

Example Questions

\tan ^{-1} \sqrt{3} = \dfrac{\pi}{3}

\arcsin \dfrac{1}{2} = \dfrac{\pi}{6}

\arccos 1 = 0

Additional Resources

MME

Exam Tips Cheat Sheet

A Level
MME

Formula Booklet

A Level

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