Answer
(a) $\cos^{-1}\Big(\frac{1}{2}\Big)=\frac{\pi}{3}$
(b) $\cos^{-1}\Big(\frac{-1}{\sqrt2}\Big)=\frac{3\pi}{4}$
(c) $\cos^{-1}\Big(\frac{\sqrt3}{2}\Big)=\frac{\pi}{6}$
Work Step by Step
*Recall the definition of arccosine:
$y=\cos^{-1}x$ is the number in $[0,\pi]$ for which $\cos y=x$
(a) $$\cos^{-1}\Big(\frac{1}{2}\Big)=a$$
According to the defintion: $$\cos a=\frac{1}{2}\hspace{1cm}\text{and}\hspace{1cm}a\in[0,\pi]$$
So $a=\frac{\pi}{3}$. In other words, $$\cos^{-1}\Big(\frac{1}{2}\Big)=\frac{\pi}{3}$$
(b) $$\cos^{-1}\Big(\frac{-1}{\sqrt2}\Big)=\cos^{-1}\Big(\frac{-\sqrt2}{2}\Big)=a$$
According to the defintion: $$\cos a=\frac{-\sqrt2}{2}\hspace{1cm}\text{and}\hspace{1cm}a\in[0,\pi]$$
So $a=\frac{3\pi}{4}$. In other words, $$\cos^{-1}\Big(\frac{-1}{\sqrt2}\Big)=\frac{3\pi}{4}$$
(c) $$\cos^{-1}\Big(\frac{\sqrt3}{2}\Big)=a$$
According to the defintion: $$\cos a=\frac{\sqrt3}{2}\hspace{1cm}\text{and}\hspace{1cm}a\in[0,\pi]$$
So $a=\frac{\pi}{6}$. In other words, $$\cos^{-1}\Big(\frac{\sqrt3}{2}\Big)=\frac{\pi}{6}$$
