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.

\[ \arcsin x \nonumber \]

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*}

\[ \arccsc x \nonumber \]

Domain: \(|x|\ge 1\)

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

\[ \arccsc x \ne 0 \nonumber \]

\[ \arcsec x \nonumber \]

Domain: \(|x|\ge 1\)

Range: \(0 \leq \arcsec x \leq \pi\)

\[ \arcsec x \ne \frac{\pi}{2} \nonumber \]

\[ \arccot x \nonumber \]

Domain: all real numbers

Range: \(0 \lt \arccot x \lt \pi\)

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*}