3. Trigonometry

e. Inverse Trig Functions

4. Inverse Secant & Cosecant

Arc Secant

The secant function is not one-to-one, since for example \[ \sec(-\theta)=\sec(\theta) \] We pick the branch with \(0 \le \theta \lt \dfrac{\pi}{2}\) or \(\dfrac{\pi}{2} \lt \theta \le \pi\).

The inverse function of \(\sec\) is \(\text{arcsec}\) (read “arc secant”) or \(\sec^{-1}\) (read “inverse secant”) which satisfies \[ \text{arcsec}(z)=\theta \qquad \text{where} \qquad z=\sec(\theta) \] provided \(|z| \ge 1\) and \(0 \le \theta \lt \dfrac{\pi}{2}\) or \(\dfrac{\pi}{2} \lt \theta \le \pi\).

Notice that \(0 \le \theta \lt \dfrac{\pi}{2}\) or \(\dfrac{\pi}{2} \lt \theta \le \pi\) is quadrants I and II only but not the \(y\)-axis.

Compute each of the following.

\(\text{arcsec}\,2\)

\(\text{arcsec}\,2=\dfrac{\pi}{3}\)
\(\text{arcsec}(-\sqrt{2})\)

\(\text{arcsec}(-\sqrt{2})=\dfrac{3\pi}{4}\)
\(\text{arcsec}\,\dfrac{2}{\sqrt{3}}\)

\(\text{arcsec}\,\dfrac{2}{\sqrt{3}}=\dfrac{\pi}{6}\)
\(\text{arcsec}(-1)\)

\(\text{arcsec}(-1)=\pi\)

Arc Cosecant

The cosecant function is not one-to-one, since for example \[ \csc(\pi-\theta)=\csc(\theta) \] We pick the branch with \(-\,\dfrac{\pi}{2} \le \theta \lt 0\) or \(0 \lt \theta \le \dfrac{\pi}{2}\).

The inverse function of \(\csc\) is \(\text{arccsc}\) (read “arc cosecant”) or \(\csc^{-1}\) (read “inverse cosecant”) which satisfies \[ \text{arccsc}(z)=\theta \qquad \text{where} \qquad z=\csc(\theta) \] provided \(|z| \ge 1\) and \(-\,\dfrac{\pi}{2} \le \theta \lt 0\) or \(0 \lt \theta \le \dfrac{\pi}{2}\).

Notice that \(-\,\dfrac{\pi}{2} \le \theta \lt 0\) or \(0 \lt \theta \le \dfrac{\pi}{2}\) is quadrants IV and I only but not the \(x\)-axis.