Inverse Sine and Cosine Functions

Definition Sine is not one-to-one on the real numbers, since for example $\sin \frac{\pi}{6} = \sin \frac{5\pi}{6} = \frac{1}{2}$. The restricted sine function is sine with domain $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. On this interval sine is increasing and takes every value in $[-1, 1]$ exactly once. The restricted cosine function is cosine with domain $[0, \pi]$, where cosine is decreasing and takes every value in $[-1, 1]$ exactly once. Both restricted functions are one-to-one, so both have inverse functions, as in inverse functions and domain restriction.

Definition When a function is not one-to-one, a branch is its restriction to a set on which it is one-to-one, usually an interval. This is a different use of the word from the branches of the tangent, cotangent, secant and cosecant graphs, which are the pieces of a graph between two consecutive vertical asymptotes. The principal branch is the agreed standard choice of such a restriction, one on which the function still takes every value of its range. For sine the principal branch is the restriction to $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, and for cosine it is the restriction to $[0, \pi]$. The inverse of the principal branch is called the principal branch of the inverse function, and its outputs are called principal values. Both intervals contain every angle between $0$ and $\frac{\pi}{2}$, so when the sine or cosine of an acute angle in a right triangle is known, the inverse returns that acute angle.

Definition For $x$ in $[-1, 1]$, the arcsine of $x$, written $\arcsin x$ or $\sin^{-1} x$, is the unique number $y$ in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ with $\sin y = x$. In symbols

Arcsine is the inverse of the restricted sine, so

It is increasing and odd, $\arcsin(-x) = -\arcsin x$ for every $x$ in $[-1, 1]$.

Definition For $x$ in $[-1, 1]$, the arccosine of $x$, written $\arccos x$ or $\cos^{-1} x$, is the unique number $y$ in $[0, \pi]$ with $\cos y = x$. In symbols

Arccosine is the inverse of the restricted cosine, so

It is decreasing, and since $\cos(\pi - y) = -\cos y$, it satisfies $\arccos(-x) = \pi - \arccos x$ for every $x$ in $[-1, 1]$. It is neither even nor odd.

Definition The domain of both $\arcsin$ and $\arccos$ is the closed interval $[-1, 1]$, the range of sine and of cosine. For a real number $x$ with $\lvert x \rvert > 1$ no real angle has sine or cosine equal to $x$, so $\arcsin x$ and $\arccos x$ are not defined as real numbers. A composite expression such as $\arcsin\bigl(u(x)\bigr)$ is defined exactly for the $x$ in the domain of $u$ with $-1 \le u(x) \le 1$.

Definition The range of $\arcsin$ is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and the range of $\arccos$ is $[0, \pi]$. The outputs are real numbers, read as angles in radians, and in degrees the ranges are $[-90^\circ, 90^\circ]$ and $[0^\circ, 180^\circ]$. So the terminal side of an arcsine value lies in Quadrant I or IV, on the positive $x$-axis or on the $y$-axis, and the terminal side of an arccosine value lies in Quadrant I or II, on the $x$-axis or on the positive $y$-axis. The two outputs are linked. For every $x$ in $[-1, 1]$

To see this, let $y = \arcsin x$. Then $\frac{\pi}{2} - y$ lies in $[0, \pi]$. Cosine is even and $\cos u = \sin\left(u + \frac{\pi}{2}\right)$ for every real $u$, so

and therefore $\frac{\pi}{2} - y = \arccos x$.

Definition The notation $\sin^{-1} x$ means $\arcsin x$, the inverse function. It does not mean the reciprocal $\frac{1}{\sin x}$, which is $\csc x$ and is written $(\sin x)^{-1}$. Likewise $\cos^{-1} x = \arccos x$, while $(\cos x)^{-1} = \sec x$. This clashes with the habit of writing $\sin^2 x$ for $(\sin x)^2$, which is one reason many books prefer the names $\arcsin$ and $\arccos$.