﻿ Inverse Trigonometric Function: Arctangent (Part 2)

# Inverse Trigonometric Function: Arctangent (Part 2)

## 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 ‘$\,\arctan\,$’.

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:

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

Range:  $\,(-\infty,\infty)\,$

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

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

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