Question

If $x+y=\frac{2\pi }{3}$ and $\frac{\sin x}{\sin y}=2$, then which of the following option(s) is/are correct?

• A. there are $4$ values of $x$ in $[0,4\pi ]$ for $y\in R-\left\{n\pi \right\}$
• B. there are $6$ values of $x$ in $[0,4\pi ]$ for $y\in R-\left\{n\pi \right\}$
• C. there are $6$ values of $y$ in $[0,4\pi ]$ for $x\in R$
• D. there are $4$ values of $y$ in $[0,4\pi ]$ for $x\in R$

Answer 1

Answer: (A), (D)

The correct options are
A there are $4$ values of $x$ in $[0,4\pi ]$ for $y\in R-\left\{n\pi \right\}$
D there are $4$ values of $y$ in $[0,4\pi ]$ for $x\in R$

$x+y=\frac{2\pi }{3}\Rightarrow y=\frac{2\pi }{3}-x$
$\therefore \sin x=2\sin (\frac{2\pi }{3}-x)$
$\Rightarrow \sin x=2[\left(\frac{\sqrt{3}}{2}\right)\cos x+\frac{1}{2}\sin x]\Rightarrow \sin x=\sqrt{3}\cos x+\sin x\Rightarrow \cos x=0\Rightarrow x=\frac{(2n+1)\pi }{2},n\in Z$
$\Rightarrow y=\frac{2\pi }{3}-n\pi -\frac{\pi }{2}$
$\Rightarrow y=\frac{\pi }{6}-n\pi ,n\in Z$

Hence, for $x\in [0,4\pi ],$
$x=\frac{\pi }{2},\frac{3\pi }{2},\frac{5\pi }{2},\frac{7\pi }{2}$

and for $y\in [0,4\pi ],$
$y=\frac{\pi }{6},\frac{7\pi }{6},\frac{13\pi }{6},\frac{19\pi }{6}$