# A.9: Inverse Trigonometric Functions

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Some of you may not have studied inverse trigonometric functions in highschool, however we still expect you to know them by the end of the course.

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*}

Again

\begin{align*} \textrm{arccsc}\,(\csc \theta) &= \theta & -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2},\ \theta\ne 0\\ \textrm{arcsec}\,(\sec \theta) & = \theta & 0 \leq \theta \leq \pi,\ \theta\ne \frac{\pi}{2}\\ \textrm{arccot}\,(\cot \theta) & = \theta & 0 \lt \theta \lt \pi \end{align*}

and

\begin{align*} \csc(\textrm{arccsc}\, x) &= x & |x|\ge 1\\ \sec(\textrm{arcsec}\, x) &= x & |x|\ge 1\\ \cot(\textrm{arccot}\, x) &= x & \text{any real } x \end{align*}

This page titled A.9: 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.