Reduction and Double-Angle Given , find .

Trigonometry Solution – Problem 7: find

Reduction and Double-Angle

Given $\sin\left(\alpha - \frac{\pi}{6}\right) = \frac{1}{4}$ , find $\sin\left(2\alpha + \frac{5\pi}{6}\right)$ .

Transform the angle:

$\sin\left(2\alpha + \frac{5\pi}{6}\right) = \sin\left[2\left(\alpha - \frac{\pi}{6}\right) + \frac{\pi}{2} + \pi\right]$

$= -\sin\left[2\left(\alpha - \frac{\pi}{6}\right) + \frac{\pi}{2}\right]$

$= -\cos\left[2\left(\alpha - \frac{\pi}{6}\right)\right]$

$= -\left(1 - 2\sin^2\left(\alpha - \frac{\pi}{6}\right)\right)$

$= -1 + 2 \cdot \left(\frac{1}{4}\right)^2 = -1 + \frac{1}{8} = \frac{7}{8}$

$\frac{7}{8}$

Correct identity setup (2 pts): choose an appropriate sum/difference, double-angle, or auxiliary-angle idea and set up the key equation(s).
Correct algebra / trig simplification (2 pts): transform expressions without sign mistakes.
Solve for target quantity (2 pts): isolate the requested value and handle any constraints if needed.
Final answer (1 pt): clearly state the result in the required form.