# 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

## 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 Level

## 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 Level

## Example Questions

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

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

$\arccos 1 = 0$

A Level

A Level

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