Question

If $si{n}^{-1}x+si{n}^{-1}y=\frac{2\pi }{3}$ and $co{s}^{-1}x-co{s}^{-1}y=\frac{\pi }{3}$ Then, $(x,y)$ is equal to

• A. $(0,1)$
• B. $(1/2,1)$
• C. $(1,1/2)$
• D. $(\sqrt{3}/2,1)$

Answer 1

Answer: (B)

We have,
${\sin }^{-1}x+{\sin }^{-1}y=\frac{2\pi }{3}....(i)$
$\left(\frac{\pi }{2}-{\sin }^{-1}x\right)-\left(\frac{\pi }{2}-{\sin }^{-1}y\right)=\frac{\pi }{3}$
$\Rightarrow -{\sin }^{-1}x+{\sin }^{-1}y=\frac{\pi }{3}...(ii)$
On adding Eqs. (i) and (ii),
${\sin }^{-1}y=\frac{\pi }{2}\Rightarrow y=1$
On subtracting Eq. (i) from Eq. (ii),
${\sin }^{-1}x=\frac{\pi }{6}\Rightarrow x=\frac{1}{2}$
$\therefore (x,y)=\left(\frac{1}{2},1\right)$