Question

Evaluate each of the following:
(i) ${\cos }^{-1}\left\{\cos \left(-\frac{\mathrm{\pi }}{4}\right)\right\}$
(ii) ${\cos }^{-1}\left\{\cos \frac{5\mathrm{\pi }}{4}\right\}$
(iii) ${\cos }^{-1}\left\{\cos \left(\frac{4\mathrm{\pi }}{3}\right)\right\}$
(iv) ${\cos }^{-1}\left(\cos \frac{13\mathrm{\pi }}{6}\right)$
(v) ${\cos }^{-1}\left(\cos 3\right)$

(vi) ${\cos }^{-1}\left(\cos 4\right)$

(vii) ${\cos }^{-1}\left(\cos 5\right)$

(viii) ${\cos }^{-1}\left(\cos 12\right)$

Answer 1

We know

${\cos }^{-1}\left(\cos \theta \right)=\theta \mathrm{if}0\le \theta \le \mathrm{\pi }$

(i) We have

$$\begin{gathered} {\cos }^{-1}\left\{\cos \left(-\frac{\mathrm{\pi }}{4}\right)\right\}={\cos }^{-1}\left\{\cos \left(\frac{\mathrm{\pi }}{4}\right)\right\} \\ =\frac{\mathrm{\pi }}{4} \end{gathered}$$

(ii) We have

$$\begin{gathered} {\cos }^{-1}\left\{\cos \left(\frac{5\mathrm{\pi }}{4}\right)\right\}={\cos }^{-1}\left\{\cos \left(2\mathrm{\pi }-\frac{3\mathrm{\pi }}{4}\right)\right\} \\ ={\cos }^{-1}\left\{\cos \left(\frac{3\mathrm{\pi }}{4}\right)\right\} \\ =\frac{3\mathrm{\pi }}{4} \end{gathered}$$

(iii) We have

$$\begin{gathered} {\cos }^{-1}\left\{\cos \left(\frac{4\mathrm{\pi }}{3}\right)\right\}={\cos }^{-1}\left\{\cos \left(2\mathrm{\pi }-\frac{2\mathrm{\pi }}{3}\right)\right\} \\ ={\cos }^{-1}\left\{\cos \left(\frac{2\mathrm{\pi }}{3}\right)\right\} \\ =\frac{2\mathrm{\pi }}{3} \end{gathered}$$

(iv) We have

$$\begin{gathered} {\cos }^{-1}\left\{\cos \left(\frac{13\mathrm{\pi }}{6}\right)\right\}={\cos }^{-1}\left\{\cos \left(2\mathrm{\pi }+\frac{\mathrm{\pi }}{6}\right)\right\} \\ ={\cos }^{-1}\left\{\cos \left(\frac{\mathrm{\pi }}{6}\right)\right\} \\ =\frac{\mathrm{\pi }}{6} \end{gathered}$$

(v) We have

${\cos }^{-1}\left(\cos 3\right)=3$

(vi)We have

$$\begin{gathered} {\cos }^{-1}\left(\cos 4\right)={\cos }^{-1}\left\{\cos \left(2\mathrm{\pi }-4\right)\right\} \\ =2\mathrm{\pi }-4 \end{gathered}$$

(vii) We have

$$\begin{gathered} {\cos }^{-1}\left(\cos 5\right)={\cos }^{-1}\left\{\cos \left(2\mathrm{\pi }-5\right)\right\} \\ =2\mathrm{\pi }-5 \end{gathered}$$

(viii) We have

$$\begin{gathered} {\cos }^{-1}\left(\cos 12\right)={\cos }^{-1}\left\{\cos \left(4\mathrm{\pi }-12\right)\right\} \\ =4\mathrm{\pi }-12 \end{gathered}$$