Question

If $x={\log }_{0.75}\left(\frac{148}{111}\right)$ and $y=\frac{\cos (\frac{\pi }{4}-\theta )-\cos (\frac{\pi }{4}+\theta )}{\sin (\frac{2\pi }{3}+\theta )-\sin (\frac{2\pi }{3}-\theta )}$ and $x-y=\tan \alpha$ then find $\alpha .$

Answer 1

$\frac{148}{111}=\frac{4}{3}$

$\therefore$ $x={log}_{3/4}4/3=-1$

$y=\frac{2\left[\cos \pi /4\cos \theta \right]}{2\left[\sin 2\pi /3\cos \theta \right]}=\frac{\frac{1}{\sqrt{2}}}{\frac{\sqrt{3}}{2}}=\sqrt{\frac{2}{3}}$

$x-y=-1-\sqrt{2/3}=\tan \alpha$

$\therefore$ $\alpha ={\tan }^{-1}(-1-\sqrt{2/3})=-61.{167}^0$