Inverse Trigonometric Function: Arccosine

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

This section is a copy of Inverse Trigonometric Function: Arcsine, with appropriate changes.
If you've mastered the arcsine section, then this one should be quick and easy!

The arccosine function was informally introduced in Using the Law of Cosines in the SSS case (& Introduction to the Arccosine Function).
It is made precise here.

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

So, to try and define an ‘inverse cosine function’, we do the best we can.
We throw away most of the cosine 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 cosine function (the interval $\,[-1,1]\,$)
  • the piece is close to the origin
The cosine curve has two pieces that meet all three requirements:
  • the green piece shown at right
  • the red piece shown at right
The green piece is nicer, because it has positive $x$-values.
Positive numbers are easier to work with than negative numbers.

This green piece is the restriction of the cosine curve to the interval $\,[0,\pi]\,$.

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

the cosine function doesn't have a true inverse.

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



The cosine 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 cosine 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 cosine function’.

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

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

The cosine 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 cosine function.

When we try to use the cosine 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 cosine 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 $\,[0,\pi]\,$.

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

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

$\,\arccos 0.5\,$ is the number in the interval $\,[0,\pi]\,$ whose cosine is $\,0.5\,$

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

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

$\,\arccos x\,$ is the number in the interval $\,[0,\pi]\,$ whose cosine is $\,x\,$

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

$\,\arccos x\,$ is the number between $\,0\,$ and $\,\pi\,$ whose cosine 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:
$\,\arccos x\,$ is the number between $\,0\,$ and $\,\pi\,$ (including the endpoints) whose cosine is $\,x\,$
... but then it loses its simplicity. Ah—issues with language. Choose words that work for you!

Precise Definition of the Arccosine Function

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

DEFINITION the arccosine function, denoted by  $\,\arccos\,$  or  $\,\cos^{-1}\,$
Let $\,-1 \le x\le 1\,$.

Using the notation ‘$\,\arccos\,$’ for the arccosine function: $$ y = \arccos x\ \ \ \ \ \text{if and only if}\ \ \ \ \bigl(\ \cos y = x\ \ \text{AND}\ \ 0 \le y\le \pi\ \bigr) $$ Using the notation ‘$\,\cos^{-1}\,$’ for the arccosine function: $$ y = \cos^{-1} x\ \ \ \ \ \text{if and only if}\ \ \ \ \bigl(\ \cos y = x\ \ \text{AND}\ \ 0 \le y\le \pi\ \bigr) $$

Notes on the Definition of the Arccosine Function:

Example: Find the Exact Value of $\,\arccos(-0.5)\,$
(use both the unit circle and a special triangle)

  • Using the degree definition:
    $\,\arccos (-\frac 12)\,$ is the angle between
    $\,0^\circ\,$ and $\,180^\circ\,$ whose cosine is $\,-\frac 12\,$.
  • Recall: cosine is the $x$-value of points on the unit circle.
  • Draw a unit circle.
    Mark $\,-\frac 12\,$ on the $x$-axis.
    Mark the unique angle between $\,0^\circ\,$ and $\,180^\circ\,$ that has this cosine value.
    This (positive) angle is $\,\arccos(-\frac 12)\,$.
  • Does any special triangle tell us an acute angle whose cosine is $\,\frac12\,$?
    Yes! The cosine of $\,60^\circ\,$ is $\,\frac 12\,$.
    Thus, the reference angle is $\,\color{red}{60^\circ}\,$,
    as shown.
  • Thus, $\,\arccos(-\frac 12) = 180^\circ - 60^\circ = 120^\circ\,$.
  • Using radian measure: $$\displaystyle\,\arccos(-\frac 12) = \frac{2\pi}3\,$$
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: Arctangent


On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
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