# Inverse Trigonometric Function: Arcsine (Part 2)

(This page is Part 2. Click here for Part 1.)

## Formal Name Versus Function Notation Abbreviations

The formal name of the function being discussed in this section is ‘arcsine’. It has an ‘e’ on the end, and is pronounced ARC-sine (with a long ‘i’ in ‘sine’).

When using function notation, ‘arcsine’ is abbreviated as ‘$\,\arcsin\,$’. It has no ‘e’ on the end, but is pronounced the same as ‘arcsine’. Thus, ‘$\,\arcsin x\,$’ is read aloud as ‘arcsine of $\,x\,$’.

An alternative notation for the arcsine function is ‘$\,\sin^{-1}\,$’. This alternative notation is modeled on the standard notation for inverse functions: if $\,f\,$ is one-to-one, then its inverse is called $\,f^{-1}\,.$

The function notation ‘$\,\sin^{-1} x\,$’ can be read aloud as ‘arcsine of $\,x\,$’ or ‘the inverse sine of $\,x\,$’. DON'T read ‘$\,\sin^{-1} x\,$’ as ‘sine to the negative one of $\,x\,$’! There is no reciprocal operation going on here—it's just standard notation for an inverse function.

## Convention for Multi-Letter Function Names

Since both ‘$\,\arcsin\,$’ and ‘$\,\sin^{-1}\,$’ are multi-letter function names, the standard convention applies: the parentheses that typically hold the input can be removed, if there is no possible confusion about order of operations.

Thus, you usually see ‘$\,\arcsin x\,$’ and ‘$\,\sin^{-1} x\,$’ (no parentheses), instead of the more cumbersome ‘$\,\arcsin (x)\,$’ and ‘$\,\sin^{-1} (x)\,$’ (with parentheses).

## Function Name Versus Output From the Function

The function name is
‘$\,\arcsin\,$’.
The number
‘$\,\arcsin x\,$’
is the *output* from the function
‘$\,\arcsin\,$’
when the input is $\,x\,.$

Similarly, the function name is
‘$\,\sin^{-1}\,$’.
The number
‘$\,\sin^{-1} x\,$’
is the *output* from the function
‘$\,\sin^{-1}\,$’
when the input is $\,x\,.$

## Preferred Notation

Since the sine function does *not*
have a true inverse,
this author believes the notation
‘$\,\sin^{-1}\,$’
is misleading and lends
itself to errors.
This author strongly prefers the notation
‘$\,\arcsin\,$’.

## Alternate Definition, Using Degrees Instead of Radians

Inputs to trigonometric functions can be viewed as real numbers (radian measure) or degrees.

For example, $\,\sin \frac{\pi}{2} = 1\,$: here, $\,\frac{\pi}{2}\,$ is radian measure.

Equivalently, $\,\sin 90^\circ = 1\,$: here, $\,90^\circ\,$ is degree measure.

Here's what the definition of arcsine looks like, using degree measure instead of radian measure:

$$ \begin{gather} \cssId{s31}{y = \arcsin x}\cr\cr \cssId{s32}{\text{if and only if}}\cr\cr \cssId{s33}{\bigl(\ \sin y = x\ \ \text{and}\ \ -90^\circ \le y\le 90^\circ\ \bigr)} \end{gather} $$## Calculator Skills

If a calculator is in degree mode, then $\,\arcsin x\,$ is reported in degrees. If a calculator is in radian mode, then $\,\arcsin x\,$ is reported in radians.

## Graph of the Arcsine Function

For a one-to-one function $\,f\,,$ the graph of its inverse, $\,f^{-1}\,,$ is found by reflecting the graph of $\,f\,$ about the line $\,y = x\,.$ Below, this technique is used to construct the graph of the arcsine function:

Domain: $\,[-\frac{\pi}{2},\frac{\pi}{2}]$

Range: $\,[-1,1]$

Domain: $\,[-1,1]$

Range: $\,[-\frac{\pi}{2},\frac{\pi}{2}]$

Notice that the domain and range of a function and its inverse are switched! The domain of one is the range of the other. The range of one is the domain of the other.

## Relationship Between the Sine and Arcsine Functions

For a one-to-one function $\,f\,$ and its inverse $\,f^{-1}\,,$ there is a simple ‘undoing’ relationship between the two:

- $f^{-1}\bigl(f(x)\bigr) = x\,$ for all $\,x\,$ in the domain of $\,f\,$: the function $\,f\,$ does something, and $\,f^{-1}\,$ undoes it
- $f\bigl(f^{-1}(x)\bigr) = x\,$ for all $\,x\,$ in the range of $\,f\,$: the function $\,f^{-1}\,$ does something, and $\,f\,$ undoes it

Since the sine and arcsine functions
are *not* true inverses of each other,
the relationship between them
is a bit more complicated.

## The Direction Where Sine and Arcsine ‘Undo’ Each Other Nicely

Here's the direction where they
*do* ‘undo’ each other nicely:
Start with a number, first apply the arcsine function,
then apply the sine function,
and end up right where you started.

Here are the details: For all $\,x\in [-1,1]\,,$

$$ \cssId{s67}{\sin(\arcsin x) = x} $$- Start with $\,\color{red}{x}\in [-1,1]$
- The arcsine function takes $\,\color{red}{x}\,$ to $\,\color{green}{\arcsin x}\,$ in the interval $\,[-\frac{\pi}2,\frac{\pi}2]$
- The sine function takes $\,\color{green}{\arcsin x}\,$ back to $\,\color{red}{x}$

## The Direction Where Sine and Arcsine Don't Necessarily ‘Undo’ Each Other Nicely

Here's the direction where they
*don't necessarily* ‘undo’ each other nicely.
Start with a number, first apply the sine function,
then apply the arcsine function.
If the number you started with is outside
the interval $\,[-\frac{\pi}2,\frac{\pi}2]\,,$
then you don't end up where you started!

Here are the details:

For all $\,x\in [-\frac{\pi}2,\frac{\pi}2]\,,$

$$ \cssId{s77}{\arcsin(\sin x) = x} $$(See the top graph above.)

For all $\,x\not\in [-\frac{\pi}2,\frac{\pi}2]\,,$

$$ \cssId{s80}{\arcsin(\sin x) \ne x} $$(See the bottom graph above.)

## Example: Find the Exact Value of $\,\arcsin(-0.5)\,$

Use both the unit circle and a special triangle.

Using the degree definition: $\,\arcsin (-\frac 12)\,$ is the angle between $\,-90^\circ\,$ and $\,90^\circ\,$ whose sine is $\,-\frac 12\,.$

Recall: Sine is the $y$-value of points on the unit circle.

Draw a unit circle. Mark $\,-\frac 12\,$ on the $y$-axis. Mark the unique angle between $\,-90^\circ\,$ and $\,90^\circ\,$ that has this sine value. This (negative) angle is $\,\arcsin(-\frac 12)\,.$

Does any special triangle tell us an acute angle whose sine is $\,\frac12\,$? Yes! The sine of $\,30^\circ\,$ is $\,\frac 12\,.$

Thus, $\,\arcsin(-\frac 12) = -30^\circ\,.$

Using radian measure, $\,\arcsin(-\frac 12) = -\frac{\pi}6\,.$