Trigonometry – Problem 10: find

Value Finding Comprehensive

Given $\sin\alpha + \sin\left(\alpha + \frac{\pi}{3}\right) = \frac{\sqrt{3}}{3}$, find $\cos\left(2\alpha + \frac{\pi}{3}\right)$.

Expanding:

$$\sin\alpha + \sin\alpha\cos\frac{\pi}{3} + \cos\alpha\sin\frac{\pi}{3} = \frac{\sqrt{3}}{3}$$

$$\frac{3}{2}\sin\alpha + \frac{\sqrt{3}}{2}\cos\alpha = \frac{\sqrt{3}}{3}$$

$$\sqrt{3}\sin\left(\alpha + \frac{\pi}{6}\right) = \frac{\sqrt{3}}{3}$$

$$\sin\left(\alpha + \frac{\pi}{6}\right) = \frac{1}{3}$$

Therefore:

$$\cos\left(2\alpha + \frac{\pi}{3}\right) = 1 - 2\sin^2\left(\alpha + \frac{\pi}{6}\right) = 1 - \frac{2}{9} = \frac{7}{9}$$

$\frac{7}{9}$