Question

$\sin \left(\frac{\pi }{10}\right)\sin \left(\frac{3\pi }{10}\right)=$

• A. $\frac{1}{2}$
• B. $-\frac{1}{2}$
• C. $\frac{1}{4}$
• D. $1$

Answer 1

The correct option is C

$\frac{1}{4}$

Explanation for the correct answer:

Simplifying the given expression using trigonometric identities:

We know that $\frac{\pi }{10}=\left(\frac{180}{\mathrm{\pi }}\times \frac{\pi }{10}\right)°=18°\text{and}\frac{3\pi }{10}=\left(\frac{180}{\mathrm{\pi }}\times \frac{3\pi }{10}\right)°=54°$

Also, $\sin 18°=\frac{\sqrt{5}–1}{4},\cos 36°=\frac{\sqrt{5}+1}{4}$

Now, applying this in the given expression, we get

$$\begin{gathered} \sin \left(\frac{\pi }{10}\right)\sin \left(\frac{3\pi }{10}\right)=\sin 18°\sin 54° \\ =\sin 18°\sin (90°–36°) \\ =\sin 18°\cos 36°\left[\because \sin (90°-\theta )=\cos \theta \right] \\ =\left(\frac{\sqrt{5}–1}{4}\right)\left(\frac{\sqrt{5}+1}{4}\right) \\ =\frac{5-1}{16} \\ =\frac{4}{16} \\ =\frac{1}{4} \end{gathered}$$

Thus, $\sin \left(\frac{\pi }{10}\right)\sin \left(\frac{3\pi }{10}\right)=$$\frac{1}{4}$

Therefore, option (C) is the correct answer.