Question

If $si{n}^{-1}(x)+si{n}^{-1}(2x)=\frac{\pi }{3}$, then $x=$

• A. $\frac{3}{28}$
• B. $\frac{\sqrt{3}}{28}$
• C. $\sqrt{\frac{3}{28}}$
• D. $\frac{3}{\sqrt{28}}$

Answer 1

The correct option is C $\sqrt{\frac{3}{28}}$

Given $si{n}^{-1}x+si{n}^{-1}\left(2x\right)=\frac{\pi }{3}$
$si{n}^{-1}2x=\frac{\pi }{3}-si{n}^{-1}x$
take sin on both sides,
$2x=sin\frac{\pi }{3}cossi{n}^{-1}x-cos\frac{\pi }{3}sinsi{n}^{-1}x$
$2x=sin\frac{\pi }{3}cossi{n}^{-1}x-cos\frac{\pi }{3}sinsi{n}^{-1}x$
$2x=\frac{\sqrt{3}}{2}\sqrt{1-x^2}-\frac{1}{2}x$
$2x+\frac{x}{2}-\frac{\sqrt{2(1-x^2)}}{2}$
$\frac{5x}{2}=\frac{\sqrt{3(1-x^2)}}{2}$
$25x^2=3-3x^2$
$28x^2=3$
$x^2=\frac{3}{28}$
$x=\sqrt{\frac{3}{28}}$