# Inverse Trig Functions

## Inverse Trig Functions Revision

**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

**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}.

**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.

## Inverse Trig Functions Example Questions

**Question 1:** Give the exact value of \tan ^{-1} \sqrt{3} in radians.

**[1 mark]**

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

**Question 2:** Give the exact value of \arcsin \dfrac{1}{2} in radians.

**[1 mark]**

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

**Question ****3:** Give the exact value of \arccos 1 in radians.

**[1 mark]**

\arccos 1 = 0