Question

 If \[\sin x = \frac{4}{5}\] and \[0 < x < \frac{\pi}{2}\]

, find the value of sin 4x.

 

 

Answer 1

\[\sin x = \frac{4}{5}\] and  \[0 < x < \frac{\pi}{2}\] .

\[\therefore \sin x = \sqrt{1 - \cos^2 x}\]
\[ \Rightarrow \left( \frac{4}{5} \right)^2 = 1 - \cos^2 x\]
\[ \Rightarrow \frac{16}{25} - 1 = - \cos^2 x\]
\[ \Rightarrow \frac{9}{25} = \cos^2 x\]
\[ \Rightarrow \cos x = \pm \frac{3}{5}\]

Since x lies in the 1st quadrant, cos x is positive.
Thus,

\[\cos x = \frac{3}{5}\]

Now,

\[\sin \left( 4x \right) = 2 \sin \left( 2x \right) \cos\left( 2x \right)\]
\[ = 2\left( 2 \sin x \cos x \right)\left( 1 - 2 \sin^2 x \right)\]
\[ = 2\left( 2 \times \frac{4}{5} \times \frac{3}{5} \right)\left( 1 - 2 \left( \frac{4}{5} \right)^2 \right)\]
\[ = 2\left( \frac{24}{25} \right)\left( 1 - \frac{32}{25} \right)\]
\[ = 2\left( \frac{24}{25} \right)\left( \frac{25 - 32}{25} \right)\]
\[ = 2\left( \frac{24}{25} \right)\left( \frac{- 7}{25} \right)\]
\[ = - \frac{336}{625}\]

Hence, the value of sin 4x is \[- \frac{336}{625}\] .