Inverse tangent function

Definition

The inverse tangent function, denoted as $\text{tan}^{-1}(x)$ or $\arctan(x)$, returns the angle whose tangent is $x$. It is the inverse operation of the tangent function and has a range of $-\frac{\pi}{2}$ to $\frac{\pi}{2}$.

5 Must Know Facts For Your Next Test

1. The inverse tangent function is defined for all real numbers.
2. $\arctan(1) = \frac{\pi}{4}$ because $\tan(\frac{\pi}{4}) = 1$.
3. The graph of $y = \arctan(x)$ is an odd function, symmetric about the origin.
4. $y = \arctan(x)$ has horizontal asymptotes at $y = \pm \frac{\pi}{2}$.
5. In solving trigonometric equations, the principal value of $\arctan(x)$ lies within the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.

Review Questions

• What is the range of the inverse tangent function?
• Calculate $\arctan(0)$. What angle does this correspond to?
• Explain why $y = \arctan(x)$ has horizontal asymptotes at $ y= \pm \frac{\pi}{2}$.

Related terms

Tangent Function: A trigonometric function that relates an angle in a right triangle to the ratio of the opposite side to the adjacent side.

Inverse Sine Function: $$ \text {denoted as } \sin^{-1}(x) \text { or } \arcsin(x), \text {returns the angle whose sine is } x. \text {Its range is } -\frac{\pi}{2} \text { to } \frac{\pi}{2}. $$

Inverse Cosine Function: $$ \text {denoted as } \cos^{-1}(x) \text { or } \arccos(x), \text {returns the angle whose cosine is } x. \text {Its range is } 0 \text { to } \pi. $$