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Inverse trigonometric functions

Inverse trigonometric functions

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Explainer Video

Tutor: Toby

Summary

Inverse trigonometric functions

​​In a nutshell

Often you will use the inverse trigonometric functions. But it is also important to understand their graphs and their domains and ranges for solving problems.



Names

The inverse trigonometric functions can be referred to in different ways. They can either be called "inverse [trigonometric function]" or "arc[trigonometric function]". So for example the inverse of sine is referred to as "sine inverse" or "arcsine". On the calculator, this is sin⁡−1\sin^{-1}sin−1.



Graphs

The graphs of y=arcsin⁡(x)y=\arcsin(x)y=arcsin(x), y=arccos⁡(x)y=\arccos(x)y=arccos(x) and y=arctan⁡(x)y=\arctan(x)y=arctan(x) can be sketched by respectively reflecting the curves y=sin⁡(x)y=\sin(x)y=sin(x), y=cos⁡(x)y=\cos(x)y=cos(x) and y=tan⁡(x)y=\tan(x)y=tan(x) in the line y=xy=xy=x. 


Arcsine

Maths; Trigonometric functions; KS5 Year 13; Inverse trigonometric functions


This is the graph of y=arcsin⁡(x)y=\arcsin(x)y=arcsin(x), in radians. Its domain is [−1,1][-1,1][−1,1] and its range is [−π2,π2]\left[-\frac{\pi}2,\frac{\pi}2\right][−2π​,2π​] (or if in degrees, [−90,90][-90,90][−90,90]). Note: This is not a true reflection of the whole of y=sin⁡(x)y=\sin(x)y=sin(x) in the line y=xy=xy=x​. If it was, then what you would get would not be a function. Instead you would have a one-to-many mapping.

​

Arccosine

Maths; Trigonometric functions; KS5 Year 13; Inverse trigonometric functions

​

This is the graph of y=arccos⁡(x)y=\arccos(x)y=arccos(x) in radians. Its domain is also [−1,1][-1,1][−1,1] and its range is [0,π][0,\pi][0,π] (or if in degrees, [0,180][0,180][0,180]). As with the arcsine graph, in order to be a function, it is not a full reflection of y=cos⁡(x)y=\cos(x)y=cos(x) in the line y=xy=xy=x.

​

Arctangent

Maths; Trigonometric functions; KS5 Year 13; Inverse trigonometric functions


This is the graph of y=arctan⁡(x)y=\arctan(x)y=arctan(x). Its domain is all real numbers and its range is (−π2,π2)\left(-\frac{\pi}2,\frac{\pi}2\right)(−2π​,2π​) (or in degrees, (−90,90)(-90,90)(−90,90)). Notice that the interval boundaries are not included in the range. This is shown by the horizontal asymptotes on the graph.


Example 1

Work out in radians, the value of arccos⁡(0.5)\arccos(0.5)arccos(0.5).


This is the same as solving the equation cos⁡(x)=0.5\cos(x)=0.5cos(x)=0.5​ in the interval [0,π][0,\pi][0,π]. Using the cos⁡−1\cos^{-1}cos−1 button on the calculator, you have 

arccos⁡(0.5)=cos⁡−1(0.5)=π3‾\begin{aligned}\arccos(0.5)&=\cos^{-1}(0.5)\\&=\underline{\frac{\pi}3}\end{aligned}arccos(0.5)​=cos−1(0.5)=3π​​​​​


Example 2

Without using a calculator, find the value of arcsin⁡(−0.5)\arcsin(-0.5)arcsin(−0.5).


This is the same as solving sin⁡(x)=−0.5\sin(x)=-0.5sin(x)=−0.5 in the interval [−π2,π2]\left[-\frac{\pi}2,\frac{\pi}2\right][−2π​,2π​]. Use the graph of y=sin⁡(x)y=\sin(x)y=sin(x) in the interval [−π2,π2]\left[-\frac{\pi}2,\frac{\pi}2\right][−2π​,2π​]​:

Maths; Trigonometric functions; KS5 Year 13; Inverse trigonometric functions

You can see that by finding the aaa such that sin⁡(a)=0.5\sin(a)=0.5sin(a)=0.5, you will have the negative of arcsin⁡(−0.5)\arcsin(-0.5)arcsin(−0.5). In other words, arcsin⁡(−0.5)=−a\arcsin(-0.5)=-aarcsin(−0.5)=−a.


Using your known trigonometric evaluations, you have that sin⁡(π6)=0.5\sin\left(\frac{\pi}{6}\right)=0.5sin(6π​)=0.5, so a=π6a=\frac\pi6a=6π​. Thus arcsin⁡(−0.5)=−π6‾\arcsin(-0.5)=\underline{-\frac\pi6}arcsin(−0.5)=−6π​​.

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FAQs - Frequently Asked Questions

What do you call inverse trigonometric functions?

The inverse trigonometric functions can be referred to in different ways. They can either be called "inverse [trigonometric function]" or "arc[trigonometric function]". So for example the inverse of sine is referred to as "sine inverse" or "arcsine". On the calculator, this is sin^(-1).

How can you sketch the inverse trigonometric graphs?

Reflect the corresponding trigonometric graph in the line y=x.

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