Question

$co{s}^{-1}\left(44/125\right)$ is equal to

• A. $2\alpha$
• B. $3\alpha$
• C. $\pi -3\alpha$
• D. $\pi -2\alpha$

Answer 1

Answer: (C)

$\pi -3\alpha$

$\alpha =co{s}^{-1}\left(\frac{4}{5}\right)\Rightarrow cos\alpha =\frac{4}{5}\Rightarrow sin\alpha =\frac{3}{5}\Rightarrow tan\alpha =\frac{3}{4}$
$\beta =ta{n}^{-1}\frac{2}{3}\Rightarrow tan\beta =\frac{2}{3}\Rightarrow sin\beta =\frac{2}{\sqrt{13}}\Rightarrow cos\beta =\frac{3}{\sqrt{13}}$
Therefore,
$cos3\alpha =4co{s}^3\alpha -3cos\alpha =4\times \frac{64}{125}-3\times \frac{4}{5}=-\frac{44}{125}$
Hence,
$cos(\pi -3\alpha )=-3cos\alpha =\frac{44}{125}\Rightarrow co{s}^{-1}\frac{44}{125}=\pi -3\alpha$