Question

Fin an acute angle $\Theta$ , when
$\frac{cos\Theta -sin\Theta }{cos\Theta +sin\Theta }=\frac{1-\sqrt{3}}{1+\sqrt{3}}$.

Answer 1

Given $\frac{\cos \theta -\sin \theta }{\cos \theta +\sin \theta }=\frac{1-\sqrt{3}}{1+\sqrt{3}}$

$⟹\frac{1-\frac{\sin \theta }{\cos \theta }}{1+\frac{\sin \theta }{\cos \theta }}=\frac{(1-\sqrt{3}{)}^2}{(1+\sqrt{3})(1-\sqrt{3})}$

$⟹\frac{1-\tan \theta }{1+\tan \theta }=\frac{1+3-2\sqrt{3}}{1-3}$

$⟹\frac{\tan \frac{\pi }{4}-\tan \theta }{1+\tan \frac{\pi }{4}\tan \theta }=\frac{4-2\sqrt{3}}{-2}$

$⟹\tan (\frac{\pi }{4}-\theta )=\sqrt{3}-2$

$⟹\frac{\pi }{4}-\theta =-\frac{\pi }{12}$

$⟹\theta =\frac{\pi }{4}+\frac{\pi }{12}$

$⟹\theta =\frac{\pi }{3}$