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Question
If \[\sin x = \frac{4}{5}\] and \[0 < x < \frac{\pi}{2}\]
, find the value of sin 4x.
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Solution
\[ \Rightarrow \left( \frac{4}{5} \right)^2 = 1 - \cos^2 x\]
\[ \Rightarrow \frac{16}{25} - 1 = - \cos^2 x\]
\[ \Rightarrow \frac{9}{25} = \cos^2 x\]
\[ \Rightarrow \cos x = \pm \frac{3}{5}\]
Thus,
\[ = 2\left( 2 \sin x \cos x \right)\left( 1 - 2 \sin^2 x \right)\]
\[ = 2\left( 2 \times \frac{4}{5} \times \frac{3}{5} \right)\left( 1 - 2 \left( \frac{4}{5} \right)^2 \right)\]
\[ = 2\left( \frac{24}{25} \right)\left( 1 - \frac{32}{25} \right)\]
\[ = 2\left( \frac{24}{25} \right)\left( \frac{25 - 32}{25} \right)\]
\[ = 2\left( \frac{24}{25} \right)\left( \frac{- 7}{25} \right)\]
\[ = - \frac{336}{625}\]
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