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}.
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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
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