Inverse Tangent and Reciprocal Trigonometric Functions

Definition Tangent restricted to the open interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ is increasing and takes every real value exactly once. For every real number $x$, the arctangent of $x$, written $\arctan x$ or $\tan^{-1} x$, is the unique number $y$ in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ with $\tan y = x$. In symbols

Arctangent is the inverse of this restricted tangent, so

It is increasing and odd, $\arctan(-x) = -\arctan x$ for every real $x$.

Definition The domain of $\arctan$ is all real numbers, because the range of tangent is all real numbers. Its range is the open interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. The endpoints are never reached, since tangent is undefined at $\pm\frac{\pi}{2}$. Instead

so the lines $y = \frac{\pi}{2}$ and $y = -\frac{\pi}{2}$ are horizontal asymptotes of the graph of $y = \arctan x$. That graph is the reflection across the line $y = x$ of the tangent branch between the vertical asymptotes $x = -\frac{\pi}{2}$ and $x = \frac{\pi}{2}$.

Definition Cotangent restricted to the open interval $(0, \pi)$ is decreasing and takes every real value exactly once. For every real number $x$, the inverse cotangent of $x$, written $\operatorname{arccot} x$ or $\cot^{-1} x$, is the unique number $y$ in $(0, \pi)$ with $\cot y = x$. Its domain is all real numbers and its range is $(0, \pi)$. For every real $x$

The reason is that $\theta = \arctan x$ lies in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, so $\frac{\pi}{2} - \theta$ lies in $(0, \pi)$. Using $\cot u = -\tan\left(u - \frac{\pi}{2}\right)$ from graphs of tangent and cotangent and the fact that tangent is odd,

Definition Secant restricted to $\left[0, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \pi\right]$ is one-to-one and takes every value in $(-\infty, -1] \cup [1, \infty)$ exactly once. For every real $x$ with $\lvert x \rvert \ge 1$, the inverse secant of $x$, written $\operatorname{arcsec} x$ or $\sec^{-1} x$, is the unique number $y$ in $\left[0, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \pi\right]$ with $\sec y = x$. Equivalently

This works because $\sec y = x$ exactly when $\cos y = \frac{1}{x}$, and the branch is the principal branch of cosine, on $[0, \pi]$, with the single point $\frac{\pi}{2}$ removed, where cosine is $0$ and secant is undefined.

Definition Cosecant restricted to $\left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right]$ is one-to-one and takes every value in $(-\infty, -1] \cup [1, \infty)$ exactly once. For every real $x$ with $\lvert x \rvert \ge 1$, the inverse cosecant of $x$, written $\operatorname{arccsc} x$ or $\csc^{-1} x$, is the unique number $y$ in $\left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right]$ with $\csc y = x$. Equivalently

The branch is the principal branch of sine with the single point $0$ removed, where sine is $0$ and cosecant is undefined.

Definition Secant and cosecant never take values strictly between $-1$ and $1$, since $\lvert \sec y \rvert = \frac{1}{\lvert \cos y \rvert} \ge 1$ and $\lvert \csc y \rvert = \frac{1}{\lvert \sin y \rvert} \ge 1$ wherever they are defined. So the domain of $\operatorname{arcsec}$ and of $\operatorname{arccsc}$ is

while $\operatorname{arccot}$, like $\arctan$, accepts every real number. Under the reciprocal formulas, an input with $\lvert x \rvert \ge 1$ becomes $\frac{1}{x}$ in $[-1, 0) \cup (0, 1]$, which is the domain of arccosine and arcsine with $0$ removed. That is why $\operatorname{arcsec}$ never takes the value $\arccos 0 = \frac{\pi}{2}$ and $\operatorname{arccsc}$ never takes the value $\arcsin 0 = 0$. For a composite expression such as $\operatorname{arcsec}\bigl(u(x)\bigr)$, the inner value must satisfy $\lvert u(x) \rvert \ge 1$.

Definition For $\arcsin$, $\arccos$ and $\arctan$ every textbook and calculator uses the same principal ranges. For $\operatorname{arccot}$, $\operatorname{arcsec}$ and $\operatorname{arccsc}$ the ranges differ between books and software, so every statement that uses them must name its branch. This course, like the calculus course, uses the ranges in the third column of the table. The last column shows one common alternative.

With the alternative range, $\operatorname{arccot} x = \arctan \frac{1}{x}$ for $x \ne 0$. For positive inputs, and for $\operatorname{arccot} 0 = \frac{\pi}{2}$, both conventions give the same value. They can differ only for negative inputs, and they do differ at every negative input except one, since $\operatorname{arcsec}(-1) = \pi$ in both.