If sin x = 4/5 and 0 < x < π/2, Find the Value of sin 4x
Question
If \[ \sin x = \frac{4}{5} \] and \[ 0 < x < \frac{\pi}{2}, \] then find the value of \[ \sin 4x. \]
Solution
Given, \[ \sin x = \frac{4}{5} \]
Since \[ 0 < x < \frac{\pi}{2}, \] therefore \(x\) lies in the first quadrant, so cosine is positive.
Using \[ \sin^2 x + \cos^2 x = 1 \]
\[ \cos x = \sqrt{1-\sin^2 x} \]
\[ = \sqrt{1-\left(\frac{4}{5}\right)^2} \]
\[ = \sqrt{1-\frac{16}{25}} \]
\[ = \sqrt{\frac{9}{25}} \]
\[ = \frac{3}{5} \]
Now, \[ \sin 2x = 2\sin x \cos x \]
\[ = 2 \times \frac{4}{5} \times \frac{3}{5} \]
\[ = \frac{24}{25} \]
Also, \[ \cos 2x = \cos^2 x – \sin^2 x \]
\[ = \left(\frac{3}{5}\right)^2 – \left(\frac{4}{5}\right)^2 \]
\[ = \frac{9}{25} – \frac{16}{25} \]
\[ = -\frac{7}{25} \]
Using the identity \[ \sin 4x = 2\sin 2x \cos 2x \]
\[ = 2 \times \frac{24}{25} \times \left(-\frac{7}{25}\right) \]
\[ = -\frac{336}{625} \]
Final Answer
\[ \boxed{\sin 4x = -\frac{336}{625}} \]