Question

$\sin\left(\frac{\pi}{5}\right) + \sin\left(\frac{2\pi}{5}\right) + \sin\left(\frac{3\pi}{5}\right) + \sin\left(\frac{4\pi}{5}\right) =$

• A. 1
• B. $\sqrt{5}$
• C. $\frac{1}{4}(\sqrt{5} + 1)(\sqrt{10 + 2\sqrt{5}})$
• D. $\frac{1}{4}(\sqrt{5} - 1)(\sqrt{10 - 2\sqrt{5}})$

Hint:

For sums of sines at angles $\frac{k\pi}{n}$, use symmetry and sum-to-product identities, or relate to roots of unity for exact values. Simplify using known trigonometric values for specific angles.

Answer 1

The Correct Option is C

Let $S = \sin\left(\frac{\pi}{5}\right) + \sin\left(\frac{2\pi}{5}\right) + \sin\left(\frac{3\pi}{5}\right) + \sin\left(\frac{4\pi}{5}\right)$. Use the symmetry $\sin(\pi - x) = \sin x$: \[ \sin\left(\frac{3\pi}{5}\right) = \sin\left(\pi - \frac{2\pi}{5}\right) = \sin\left(\frac{2\pi}{5}\right), \quad \sin\left(\frac{4\pi}{5}\right) = \sin\left(\pi - \frac{\pi}{5}\right) = \sin\left(\frac{\pi}{5}\right) \] Thus: \[ S = \sin\left(\frac{\pi}{5}\right) + \sin\left(\frac{2\pi}{5}\right) + \sin\left(\frac{2\pi}{5}\right) + \sin\left(\frac{\pi}{5}\right) = 2 \left[ \sin\left(\frac{\pi}{5}\right) + \sin\left(\frac{2\pi}{5}\right) \right] \] Use the sum-to-product identity: \[ \sin a + \sin b = 2 \sin\left(\frac{a + b}{2}\right) \cos\left(\frac{a - b}{2}\right) \] \[ \sin\left(\frac{\pi}{5}\right) + \sin\left(\frac{2\pi}{5}\right) = 2 \sin\left(\frac{\frac{\pi}{5} + \frac{2\pi}{5}}{2}\right) \cos\left(\frac{\frac{2\pi}{5} - \frac{\pi}{5}}{2}\right) = 2 \sin\left(\frac{3\pi}{10}\right) \cos\left(\frac{\pi}{10}\right) \] \[ S = 2 \cdot 2 \sin\left(\frac{3\pi}{10}\right) \cos\left(\frac{\pi}{10}\right) = 4 \sin\left(\frac{3\pi}{10}\right) \cos\left(\frac{\pi}{10}\right) \] Use the double-angle identity: $\sin(3\theta) = 3 \sin \theta - 4 \sin^3 \theta$. For $\theta = \frac{\pi}{10}$: \[ \sin\left(\frac{3\pi}{10}\right) = 3 \sin\left(\frac{\pi}{10}\right) - 4 \sin^3\left(\frac{\pi}{10}\right) \] Thus: \[ S = 4 \left[ 3 \sin\left(\frac{\pi}{10}\right) - 4 \sin^3\left(\frac{\pi}{10}\right) \right] \cos\left(\frac{\pi}{10}\right) \] This is complex to simplify directly. Instead, use the known sum for sines over a regular pentagon’s angles, related to the fifth roots of unity. The sum $\sum_{k=1}^4 \sin\left(\frac{k\pi}{5}\right)$ is: \[ S = \frac{\sin\left(\frac{4\pi}{10}\right)}{\sin\left(\frac{\pi}{10}\right)} = \frac{\sin\left(\frac{2\pi}{5}\right)}{\sin\left(\frac{\pi}{10}\right)} \] Compute $\sin\left(\frac{\pi}{5}\right) = \frac{\sqrt{10 - 2\sqrt{5}}}{4}$, $\sin\left(\frac{\pi}{10}\right) = \frac{\sqrt{5} - 1}{4}$. The exact form is: \[ S = \frac{1}{4} (\sqrt{5} + 1) \sqrt{10 + 2\sqrt{5}} \] Option (3) is correct. Options (1), (2), and (4) do not match the computed value.