Inverse Trigonometric Function: Arcsine

Before studying this section, you are encouraged to read Trying to ‘Undo’ Trigonometric Functions.

For a function to have an inverse, each output must have exactly one corresponding input.
Thus, only one-to-one functions have inverses.
The sine function doesn't have a true inverse, because the sine function is not one-to-one.

So, to try and define an ‘inverse sine function’, we do the best we can.
We throw away most of the sine curve, leaving us with a piece that has three properties:
  • the piece is one-to-one (and hence has an inverse)
  • the piece covers all the outputs from the sine function (the interval $\,[-1,1]\,$)
  • the piece is close to the origin
The green piece shown at right satisfies all three of these properties.
This green piece is the restriction of the sine curve to the interval $\,[-\frac{\pi}{2},\frac{\pi}{2}]\,$.

The function that the mathematical community calls ‘the inverse sine function’
is not actually the inverse of the sine function, because

the sine function doesn't have a true inverse.

Instead, the ‘inverse sine function’ is the inverse of this green piece of the sine curve.
Several Cycles of the Graph of the Sine Function



The sine function isn't one-to-one;
it doesn't pass a horizontal line test.
So, it doesn't have a true inverse.

To define an ‘inverse sine function’,
we do the best we can.
Throw away most of the curve—
leave only the green part.

This green part is one-to-one.
This green part does have an inverse.

The inverse of this green part is
what the mathematical community calls
‘the inverse sine function’.

The arcsine function (precise definition below) is the best we can do in trying to get an inverse of the sine function.
The arcsine function is actually the inverse of the green piece shown above!

Here's a ‘function box’ view of what's going on:

The sine function takes a real number
as an input.
It gives an output in the interval $\,[-1,1]\,$.

For example (as below),
the output $\,0.5\,$
might come from the sine function.

When we try to use the sine function box ‘backwards’, we run into trouble.

The output $\,0.5\,$ could have come
from any of the inputs shown.

However, when we use
the green piece of the sine curve,
the problem is solved!

Now, there's only one input that works.
(It's the value of the green $\,\color{green}{x}\,$.)

Observe that $\,\color{green}{x}\,$ is in the interval $\,[-\frac{\pi}2,\frac{\pi}2]\,$.

It's a bit of a misnomer, but the arcsine function (precise definition below) is often referred to as the ‘inverse sine function’.
A better name would be something like ‘the inverse of an appropriately-restricted sine function’.
(It's no surprise, however, that people don't say something that long and cumbersome.)

So, what exactly is $\,\arcsin 0.5\,$?

$\,\arcsin 0.5\,$ is the number in the interval $\,[-\frac{\pi}{2},\frac{\pi}{2}]\,$ whose sine is $\,0.5\,$

What exactly IS $\,\arcsin x\,$?

More generally, let $\,x\,$ be any number in the interval $\,[-1,1]\,$.
Then:

$\,\arcsin x\,$ is the number in the interval $\,[-\frac{\pi}{2},\frac{\pi}{2}]\,$ whose sine is $\,x\,$

In my own mind (author Dr. Carol Burns speaking here), the words I say are:

$\,\arcsin x\,$ is the number between $\,-\frac{\pi}{2}\,$ and $\,\frac{\pi}{2}\,$ whose sine is $\,x\,$
I personally know the endpoints are included, so this doesn't confuse me.
However, the word ‘between’ is ambiguous—it can include the endpoints or not, depending on context.
It can be clarified by saying:
$\,\arcsin x\,$ is the number between $\,-\frac{\pi}{2}\,$ and $\,\frac{\pi}{2}\,$ (including the endpoints) whose sine is $\,x\,$
... but then it loses its simplicity. Ah—issues with language. Choose words that work for you!

Precise Definition of the Arcsine Function

The precise definition of the arcsine function follows.
It can look a bit intimidating—the notes following the definition should help.

DEFINITION the arcsine function, denoted by  $\,\arcsin\,$  or  $\,\sin^{-1}\,$
Let $\,-1 \le x\le 1\,$.

Using the notation ‘$\,\arcsin\,$’ for the arcsine function: $$ y = \arcsin x\ \ \ \ \ \text{if and only if}\ \ \ \ \bigl(\ \sin y = x\ \ \text{AND}\ \ -\frac{\pi}{2} \le y\le \frac{\pi}{2}\ \bigr) $$ Using the notation ‘$\,\sin^{-1}\,$’ for the arcsine function: $$ y = \sin^{-1} x\ \ \ \ \ \text{if and only if}\ \ \ \ \bigl(\ \sin y = x\ \ \text{AND}\ \ -\frac{\pi}{2} \le y\le \frac{\pi}{2}\ \bigr) $$

Notes on the Definition of the Arcsine Function:

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, $\,\arcsin(-\frac 12) = -\frac{\pi}6\,$.
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Inverse Trigonometric Function: Arccosine


On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40 41 42 43    
AVAILABLE MASTERED IN PROGRESS

(MAX is 43; there are 43 different problem types.)
Want textboxes to type in your answers? Check here: