# B.3 Inverse Trigonometric Functions

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

In order to construct inverse trigonometric functions we first have to restrict their domains so as to make them one-to-one (or injective). We do this as shown below $\sin\theta \nonumber$ Domain: $$-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$$ Range: $$-1 \leq \sin \theta \leq 1$$ $\cos \theta \nonumber$ Domain: $$0 \leq \theta \leq \pi$$ Range: $$-1 \leq \cos \theta \leq 1$$ $\tan \theta \nonumber$ Domain: $$-\frac{\pi}{2} \lt \theta \lt \frac{\pi}{2}$$ Range: all real numbers $\arcsin x \nonumber$ Domain: $$-1 \leq x \leq 1$$ Range: $$-\frac{\pi}{2} \leq \arcsin x \leq \frac{\pi}{2}$$ $\arccos x \nonumber$ Domain: $$-1 \leq x \leq 1$$ Range: $$0 \leq \arccos x \leq \pi$$ $\arctan x \nonumber$ Domain: all real numbers Range: $$-\frac{\pi}{2} \lt \arctan x \lt \frac{\pi}{2}$$

Since these functions are inverses of each other we have

\begin{align*} \arcsin(\sin \theta) &= \theta & -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\\ \arccos(\cos \theta) &= \theta & 0 \leq \theta \leq \pi\\ \arctan(\tan \theta) &= \theta & -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \end{align*}

and also

\begin{align*} \sin(\arcsin x) &= x & -1 \leq x \leq 1\\ \cos(\arccos x) &= x & -1 \leq x \leq 1\\ \tan(\arctan x) &= x & \text{any real $x$} \end{align*}

We can read other combinations of trig functions and their inverses, like, for example, $$\cos(\arcsin x)\text{,}$$ off of triangles like  We have chosen the hypotenuse and opposite sides of the triangle to be of length 1 and $$x\text{,}$$ respectively, so that $$\sin(\theta)=x\text{.}$$ That is, $$\theta = \arcsin x\text{.}$$ We can then read off of the triangle that

\begin{align*} \cos(\arcsin x) &= \cos(\theta) = \sqrt{1-x^2} \end{align*}

We can reach the same conclusion using trig identities, as follows.

• Write $$\arcsin x=\theta\text{.}$$ We know that $$\sin(\theta)=x$$ and we wish to compute $$\cos(\theta)\text{.}$$ So we just need to express $$\cos(\theta)$$ in terms of $$\sin(\theta)\text{.}$$
• To do this we make use of one of the Pythagorean identities

\begin{align*} \sin^2\theta + \cos^2\theta &=1\\ \cos\theta &= \pm \sqrt{1-\sin^2\theta} \end{align*}

• Thus

\begin{gather*} \cos(\arcsin x) = \cos\theta = \pm\sqrt{1-\sin^2\theta} \end{gather*}

• To determine which branch we should use we need to consider the domain and range of $$\arcsin x\text{:}$$

\begin{align*} \text{Domain: } -1 \leq x \leq 1 && \text{Range: } -\frac{\pi}{2} \leq \arcsin x \leq \frac{\pi}{2} \end{align*}

Thus we are applying cosine to an angle that always lies between $$-\frac{\pi}{2}$$ and $$\frac{\pi}{2}\text{.}$$ Cosine is non-negative on this range. Hence we should take the positive branch and

\begin{align*} \cos(\arcsin x) &= \sqrt{1-\sin^2\theta}= \sqrt{1-\sin^2(\arcsin x)}\\ &= \sqrt{1-x^2} \end{align*}

In a very similar way we can simplify $$\tan(\arccos x)\text{.}$$

• Write $$\arccos x=\theta\text{,}$$ and then

\begin{align*} \tan( \arccos x) &= \tan \theta = \frac{\sin\theta}{\cos \theta} \end{align*}

• Now the denominator is easy since $$\cos \theta = \cos \arccos x = x\text{.}$$
• The numerator is almost the same as the previous computation.

\begin{align*} \sin\theta &= \pm \sqrt{1-\cos^2\theta}\\ &= \pm \sqrt{1-x^2} \end{align*}

• To determine which branch we again consider domains and and ranges:

\begin{align*} \text{Domain: } -1 \leq x \leq 1 && \text{Range: } 0 \leq \arccos x \leq \pi \end{align*}

Thus we are applying sine to an angle that always lies between $$0$$ and $$\pi\text{.}$$ Sine is non-negative on this range and so we take the positive branch.
• Putting everything back together gives

\begin{align*} \tan(\arccos x) &= \frac{\sqrt{1-x^2}}{x} \end{align*}

Completing the 9 possibilities gives:

\begin{align*} \sin( \arcsin x ) &= x & \sin( \arccos x ) &= \sqrt{1-x^2} & \sin( \arctan x ) &= \frac{x}{\sqrt{1+x^2}}\\ \cos( \arcsin x ) &= \sqrt{1-x^2} & \cos( \arccos x ) &= x & \cos( \arctan x ) &= \frac{1}{\sqrt{1+x^2}}\\ \tan( \arcsin x ) &= \frac{x}{\sqrt{1-x^2}} & \tan( \arccos x ) &= \frac{\sqrt{1-x^2}}{x} & \tan( \arctan x ) &= x \end{align*}

This page titled B.3 Inverse Trigonometric Functions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Joel Feldman, Andrew Rechnitzer and Elyse Yeager via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.