Skip to main content Contents Prev Up Next \(\require{cancel}\newcommand{\dee}[1]{\mathrm{d}#1}
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Section A.9 Inverse 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: \(-1 \leq x \leq 1\)
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} \lt \theta \lt \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*}
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*}
