Find the simplified value of the following expressions $sin [\frac{π}{3} - {sin}^{- 1} (- \frac{1}{2})]$ .

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Solution

${sin}^{- 1} (\frac{- 1}{2}) = θ$ such that $sin θ = \frac{- 1}{2}$

$sin θ = \frac{- 1}{2} = - sin (\frac{π}{6}) = sin (\frac{- π}{6})$

Because $- sin θ = sin (- θ)$

$θ = \frac{- π}{6}$

i.e. ${sin}^{- 1} (\frac{- 1}{2}) = \frac{- π}{6}$

Now, Put value of ${sin}^{- 1} (\frac{- 1}{2})$ in given equation

$sin [\frac{π}{3} - \frac{- π}{6}] = sin [\frac{π}{3} + \frac{π}{6}] = sin [\frac{π}{2}] = 1$

$sin [\frac{π}{3} - {sin}^{- 1} (\frac{- 1}{2})] = 1$

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