Question

If $\sin {32}^o=k$ and $\cos x=1-2k^2;\alpha ,\beta$ are the values of x between $0^o$ and ${360}^o$ with $\alpha <\beta$, then the value in degrees of $\frac{\beta }{\alpha }=\frac{37}{k}$, then k is

Answer 1

Given $\sin {32}^o=k$
and $\cos x=1-2k^2$
$\Rightarrow \cos x=1-2{\sin }^{{32}^0}$
$\Rightarrow \cos x=\cos {64}^0$
Since, x lies between $0^o$ and ${360}^o$
$\Rightarrow x={64}^0,{296}^0$
So, $\alpha ={64}^0,\beta ={296}^0$ ($\because \alpha <\beta$ )
So, $\frac{\beta }{\alpha }=\frac{296}{64}=\frac{37}{8}$