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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: \(-1 \leq x \leq 1\)

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*}