First slide

Properties of ITF

Question

$If \sin^{- 1} \frac{1}{3} + \sin^{- 1} \frac{2}{3} = \sin^{- 1} x,$ then the value of x is

Moderate

A

$\frac{\sqrt{5} + \sqrt{2}}{9}$
B

$\frac{\sqrt{5} + 4 \sqrt{2}}{9}$
C

$\frac{\sqrt{5} + 2 \sqrt{2}}{9}$
D

None of these

Solution

$\sin^{- 1} \frac{1}{3} + \sin^{- 1} \frac{2}{3} = \sin^{- 1} x \begin{array}{l} \Rightarrow \sin^{- 1} (\frac{1}{3} \sqrt{1 - \frac{4}{9}} + \frac{2}{3} \sqrt{1 - \frac{1}{9}}) = \sin^{- 1} x \\ \Rightarrow \sin^{- 1} (\frac{1}{3} \cdot \frac{\sqrt{5}}{3} + \frac{2}{3} \cdot \frac{\sqrt{8}}{3}) = \sin^{- 1} x \\ \Rightarrow \sin^{- 1} (\frac{\sqrt{5} + 4 \sqrt{2}}{9}) = \sin^{- 1} x \\ \Rightarrow x = \frac{\sqrt{5} + 4 \sqrt{2}}{9} \end{array}$

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