Inverse Trigonometric Function: Arctangent (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 ‘arctangent’. It is pronounced ARC-tan-gent.

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

An alternative notation for the arctangent function is ‘$\,\tan^{-1}\,$’. 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 ‘$\,\tan^{-1} x\,$’ can be read aloud as ‘arctangent of $\,x\,$’ or ‘the inverse tangent of $\,x\,$’. DON'T read ‘$\,\tan^{-1} x\,$’ as ‘tangent 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 ‘$\,\arctan\,$’ and ‘$\,\tan^{-1}\,$’ 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 ‘$\,\arctan x\,$’ and ‘$\,\tan^{-1} x\,$’ (no parentheses), instead of the more cumbersome ‘$\,\arctan (x)\,$’ and ‘$\,\tan^{-1} (x)\,$’ (with parentheses).

Function Name Versus Output From the Function

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

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

Preferred Notation

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

Alternate Definition, Using Degrees Instead of Radians

Inputs to trigonometric functions can be viewed as real numbers (radian measure) or degrees. For example, $\,\tan \frac{\pi}{4} = 1\,$: here, $\,\frac{\pi}{4}\,$ is radian measure. Equivalently, $\,\tan 45^\circ = 1\,$: here, $\,45^\circ\,$ is degree measure.

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

$$ \begin{gather} \cssId{s31}{y = \arctan x}\cr\cr \cssId{s32}{\text{if and only if}}\cr\cr \cssId{s33}{\bigl(\ \tan y = x\ \ \text{and}\ \ -90^\circ \lt y\lt 90^\circ\ \bigr)} \end{gather} $$

$\arctan x\,$ is the angle in the interval $\,(-90^\circ,90^\circ)\,$ whose tangent is $\,x$

Calculator Skills

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

Graph of the Arctangent 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 arctangent function:

Here's the piece of the tangent curve that is used to define the arctangent function: piece of the tangent curve used to define the arctangent

Domain: $\,(-\frac{\pi}{2},\frac{\pi}{2})\,$

Range: $\,(-\infty,\infty)\,$

Here's the same curve, together with its reflection about the line $\,\color{red}{y = x}$ restricted tangent curve with reflection about the line y = x

The graph of the arctangent function

Domain: $\,(-\infty,\infty)\,$

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 Tangent and Arctangent 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 tangent and arctangent functions are not true inverses of each other, the relationship between them is a bit more complicated.

The Direction Where Tangent and Arctangent ‘Undo’ Each Other Nicely

the direction where the tangent and arctangent undo each other nicely

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

Here are the details: For all $\,x\in \Bbb R\,,$

$$ \cssId{s65}{\tan(\arctan x) = x} $$

Start with $\,\color{red}{x}\in \Bbb R$
The arctangent function takes $\,\color{red}{x}\,$ to $\,\color{green}{\arctan x}\,$ in the interval $\,(-\frac{\pi}2,\frac{\pi}2)$
The tangent function takes $\,\color{green}{\arctan x}\,$ back to $\,\color{red}{x}$

The Direction Where Tangent and Arctangent Don't Necessarily ‘Undo’ Each Other Nicely

in the interval from -pi/2 to pi/2, tangent and arctangent undo each other nicely

outside the interval from -pi/2 to pi/2, tangent and arctangent do NOT undo each other

Here's the direction where they don't necessarily ‘undo’ each other nicely. Start with a number, first apply the tangent function, then apply the arctangent 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 $\,x\in (-\frac{\pi}2,\frac{\pi}2)\,,$

$$ \cssId{s75}{\arctan(\tan x) = x} $$

(See the top graph above.)

For all $\,x\,$ in the domain of the tangent function but not in $\,(-\frac{\pi}2,\frac{\pi}2)\,,$

$$ \cssId{s78}{\arctan(\tan x) \ne x} $$

(See the bottom graph above.)

Example: Find the Exact Value of $\,\arctan(-1/\sqrt 3)$

Use both the unit circle and a special triangle.

Using the degree definition: $\,\arctan (-1/\sqrt 3)\,$ is the angle between $\,-90^\circ\,$ and $\,90^\circ\,$ whose tangent is $\,-\frac{1}{\sqrt 3}\,.$

As needed, review information about the size and sign of the tangent function.

Draw a unit circle. Since we want an angle between $\,-90^\circ\,$ and $\,90^\circ\,$ whose tangent is negative, the angle is in quadrant IV.

Since we want an angle whose tangent has size $\,\frac{1}{\sqrt 3} \approx 0.58\,,$ make the red segment have this length. The (negative) angle shown is therefore $\,\arctan(-1/\sqrt 3)\,.$

Does any special triangle tell us an acute angle whose tangent is $\,\frac{1}{\sqrt 3}\,$? Yes! The tangent of $\,30^\circ\,$ is $\,\frac{1}{\sqrt 3}\,.$

Thus, $\,\arctan(-1/\sqrt{3}) = -30^\circ\,.$

Using radian measure, $\,\arctan(-1/\sqrt{3}) = -\frac{\pi}6\,.$