Question

If $\alpha =si{n}^{-1}\left(\frac{4}{5}\right),\beta =co{s}^{-1}\left(\frac{5}{13}\right)$ and $x=sin(2\alpha +\beta )$ then the value of $\frac{540}{x}$ is equal to

Answer 1

$\alpha =si{n}^{-1}\left(\frac{4}{5}\right)=ta{n}^{-1}\left(\frac{4}{3}\right)$

$\beta =co{s}^{-1}\left(\frac{5}{13}\right)=ta{n}^{-1}\left(\frac{12}{5}\right)$

Hence

$2\alpha +\beta$

$=2ta{n}^{-1}\frac{4}{3}+ta{n}^{-1}\left(\frac{12}{5}\right)$

$=ta{n}^{-1}\left(\frac{4}{3}\right)+ta{n}^{-1}\left(\frac{4}{3}\right)+ta{n}^{-1}\left(\frac{12}{5}\right)$

$=ta{n}^{-1}\left(\frac{4}{3}\right)+ta{n}^{-1}\left(\frac{20+36}{15-48}\right)$ $[\because {\tan }^{-1}x+{\tan }^{-1}y={\tan }^{-1}\frac{x+y}{1-xy}]$

$=ta{n}^{-1}\left(\frac{4}{3}\right)-ta{n}^{-1}\left(\frac{56}{33}\right)$

$=ta{n}^{-1}\left(\frac{132-168}{99+224}\right)$

$=-ta{n}^{-1}\left(\frac{36}{323}\right)$

$=-si{n}^{-1}\left(\frac{36}{325}\right)$

Hence

$|x|=|-\frac{36}{325}|$

$\frac{540}{|x|}=\frac{540\times 325}{36}$

$=15\left(325\right)$

$=4875$