## SectionA.9Inverse Trigonometric Functions

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.

\begin{equation*} \arcsin x \end{equation*}
\begin{equation*} \arccos x \end{equation*}
\begin{equation*} \arctan x \end{equation*}

Domain: $-1 \leq x \leq 1$

Domain: $-1 \leq x \leq 1$

Domain: all real numbers

Range: $-\frac{\pi}{2} \leq \arcsin x \leq \frac{\pi}{2}$

Range: $0 \leq \arccos x \leq \pi$

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*}
\begin{equation*} \arccsc x \end{equation*}
\begin{equation*} \arcsec x \end{equation*}
\begin{equation*} \arccot x \end{equation*}

Domain: $|x|\ge 1$

Domain: $|x|\ge 1$

Domain: all real numbers

Range: $-\frac{\pi}{2} \leq \arccsc x \leq \frac{\pi}{2}$

Range: $0 \leq \arcsec x \leq \pi$

Range: $0 \lt \arccot x \lt \pi$

\begin{equation*} \arccsc x \ne 0 \end{equation*}
\begin{equation*} \arcsec x \ne \frac{\pi}{2} \end{equation*}

Again

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

and

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