Question

If $\frac{\sin \alpha }{\sin \beta }=\frac{\sqrt{3}}{2}$ and $\frac{\cos \alpha }{\cos \beta }=\frac{\sqrt{5}}{2},0<\alpha <\beta <\frac{\pi }{2}$, then which of the following is false

• A. $\cot \beta -1$
• B. $\tan \alpha -\frac{\sqrt{3}}{\sqrt{5}}$
• C. $B=\frac{\pi }{4}$
• D. $B=\frac{\pi }{3}$

Answer 1

Answer: (C)

$\sin \alpha =\frac{\sqrt{3}}{2}\sin \beta$
$\Rightarrow \cos \alpha =\frac{\sqrt{5}}{2}\cos \beta$
Squaring and adding, we get
$\Rightarrow 1=\frac{3}{4}{\sin }^2\beta =\frac{5}{4}\left(1-{\sin }^2\beta \right)$
$\frac{1}{2}{\sin }^2\beta =\frac{1}{4}$
$\Rightarrow {\sin }^2\beta =\frac{1}{2}$
$\beta =\frac{\pi }{4}$
$\Rightarrow \tan \alpha =\sqrt{\frac{3}{5}}$