Question

Evaluate each of the following:

(i) ${\sec }^{-1}\left(\sec \frac{\mathrm{\pi }}{3}\right)$
(ii) ${\sec }^{-1}\left(\sec \frac{2\mathrm{\pi }}{3}\right)$
(iii) ${\sec }^{-1}\left(\sec \frac{5\mathrm{\pi }}{4}\right)$
(iv) ${\sec }^{-1}\left(\sec \frac{7\mathrm{\pi }}{3}\right)$

(v) ${\sec }^{-1}\left(\sec \frac{9\mathrm{\pi }}{5}\right)$
(vi) ${\sec }^{-1}\left\{\sec \left(-\frac{7\mathrm{\pi }}{3}\right)\right\}$
(vii) ${\sec }^{-1}\left(\sec \frac{13\mathrm{\pi }}{4}\right)$
(viii) ${\sec }^{-1}\left(\sec \frac{25\mathrm{\pi }}{6}\right)$

Answer 1

We know that

${\sec }^1\left(\sec \theta \right)=\theta$, [0, π/2) ⋃ ( π/2, π]

(i) We have
${\sec }^{-1}\left(\sec \frac{\mathrm{\pi }}{3}\right)=\frac{\mathrm{\pi }}{3}$
(ii) We have
${\sec }^{-1}\left(\sec \frac{2\mathrm{\pi }}{3}\right)=\frac{2\mathrm{\pi }}{3}$
(iii) We have

$$\begin{gathered} {\sec }^{-1}\left(\sec \frac{5\mathrm{\pi }}{4}\right)={\sec }^{-1}\left[\sec \left(2\mathrm{\pi }-\frac{3\mathrm{\pi }}{4}\right)\right] \\ ={\sec }^{-1}\left[\sec \left(\frac{3\mathrm{\pi }}{4}\right)\right] \\ =\frac{3\mathrm{\pi }}{4} \end{gathered}$$
(iv)We have

$$\begin{gathered} {\sec }^{-1}\left(\sec \frac{7\mathrm{\pi }}{3}\right)={\sec }^{-1}\left[\sec \left(2\mathrm{\pi }+\frac{\mathrm{\pi }}{3}\right)\right] \\ ={\sec }^{-1}\left[\sec \left(\frac{\mathrm{\pi }}{3}\right)\right] \\ =\frac{\mathrm{\pi }}{3} \end{gathered}$$

(v)We have

$$\begin{gathered} {\sec }^{-1}\left(\sec \frac{9\mathrm{\pi }}{5}\right)={\sec }^{-1}\left[\sec \left(2\mathrm{\pi }-\frac{\mathrm{\pi }}{5}\right)\right] \\ ={\sec }^{-1}\left[\sec \left(\frac{\mathrm{\pi }}{5}\right)\right] \\ =\frac{\mathrm{\pi }}{5} \end{gathered}$$
(vi) We have

$$\begin{gathered} {\sec }^{-1}\left\{\sec \left(-\frac{7\mathrm{\pi }}{3}\right)\right\}={\sec }^{-1}\left\{\sec \left(\frac{7\mathrm{\pi }}{3}\right)\right\} \\ ={\sec }^{-1}\left[\sec \left(2\mathrm{\pi }+\frac{\mathrm{\pi }}{3}\right)\right] \\ ={\sec }^{-1}\left[\sec \left(\frac{\mathrm{\pi }}{3}\right)\right] \\ =\frac{\mathrm{\pi }}{3} \end{gathered}$$
(vii)We have

$$\begin{gathered} {\sec }^{-1}\left(\sec \frac{13\mathrm{\pi }}{4}\right)={\sec }^{-1}\left[\sec \left(4\mathrm{\pi }-\frac{3\mathrm{\pi }}{4}\right)\right] \\ ={\sec }^{-1}\left[\sec \left(\frac{3\mathrm{\pi }}{4}\right)\right] \\ =\frac{3\mathrm{\pi }}{4} \end{gathered}$$
(viii)We have

$$\begin{gathered} {\sec }^{-1}\left(\sec \frac{25\mathrm{\pi }}{6}\right)={\sec }^{-1}\left[\sec \left(4\mathrm{\pi }+\frac{\mathrm{\pi }}{6}\right)\right] \\ ={\sec }^{-1}\left[\sec \left(\frac{\mathrm{\pi }}{6}\right)\right] \\ =\frac{\mathrm{\pi }}{6} \end{gathered}$$