# Inverse Trigonometric Function: Arccosine (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 ‘arccosine’. It is pronounced arc-CO-sine (with a long ‘i’ in ‘sine’).

When using function notation, ‘arccosine’ is abbreviated as ‘$\,\arccos\,$’. It is pronounced the same as ‘arccosine’. Thus, ‘$\,\arccos x\,$’ is read aloud as ‘arccosine of $\,x\,$’.

An alternative notation for the arccosine function is ‘$\,\cos^{-1}\,$’. 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
‘$\,\cos^{-1} x\,$’
can be read aloud as
‘arccosine of $\,x\,$’
or ‘the inverse cosine
of $\,x\,$’.
*Don't* read
‘$\,\cos^{-1} x\,$’
as ‘cosine 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 ‘$\,\arccos\,$’ and ‘$\,\cos^{-1}\,$’ 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 ‘$\,\arccos x\,$’ and ‘$\,\cos^{-1} x\,$’ (no parentheses), instead of the more cumbersome ‘$\,\arccos (x)\,$’ and ‘$\,\cos^{-1} (x)\,$’ (with parentheses) .

## Function Name Versus Output From the Function

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

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

## Preferred Notation

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

## Alternate Definition, Using Degrees Instead of Radians

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

For example, $\,\cos \pi = -1\,$: here, $\,\pi\,$ is radian measure.

Equivalently, $\,\cos 180^\circ = -1\,$: here, $\,180^\circ\,$ is degree measure.

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

$$ \begin{gather} \cssId{s31}{y = \arccos x}\cr\cr \cssId{s32}{\text{if and only if}}\cr\cr \cssId{s33}{\bigl(\ \cos y = x\ \ \text{and}\ \ 0^\circ \le y\le 180^\circ\ \bigr)} \end{gather} $$## Calculator Skills

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

## Graph of the Arccosine 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 arccosine function:

Domain: $\,[0,\pi]$

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

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

Range: $\,[0,\pi]$

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 Cosine and Arccosine 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 cosine and arccosine functions
are *not* true inverses of each other,
the relationship between them is a bit
more complicated.

## The Direction Where Cosine and Arccosine ‘Undo’ Each Other Nicely

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

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

$$ \cssId{s66}{\cos(\arccos x) = x} $$- Start with $\,\color{red}{x}\in [-1,1]$
- The arccosine function takes $\,\color{red}{x}\,$ to $\,\color{green}{\arccos x}\,$ in the interval $\,[0,\pi]$
- The cosine function takes $\,\color{green}{\arccos x}\,$ back to $\,\color{red}{x}$

## The Direction Where Cosine and Arccosine 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 cosine function,
then apply the arccosine function.
If the number you started with is
outside the interval $\,[0,\pi]\,,$
then you don't end up where you started!

Here are the details:

For all $\,x\in [0,\pi]\,,$

$$ \cssId{s76}{\arccos(\cos x) = x} $$(See the top graph above.)

For all $\,x\not\in [0,\pi]\,,$

$$ \cssId{s79}{\arccos(\cos x) \ne x} $$(See the bottom graph above.)

## 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: $\displaystyle\,\cssId{s98}{\arccos(-\frac 12) = \frac{2\pi}3}\,.$