Question

1. find the exact value using half - a

(a) \\( \sin \left( \frac { 11 \pi } { 12 } \
ight) \\)

Explanation:

Step1: Determine the formula

Use the half - angle formula \(\sin\frac{\alpha}{2}=\pm\sqrt{\frac{1 - \cos\alpha}{2}}\). For \(\sin(\frac{11\pi}{12})\), we can write \(\frac{11\pi}{12}=\frac{\frac{11\pi}{6}}{2}\).

Step2: Find the cosine value

First, find \(\cos(\frac{11\pi}{6})\). We know that \(\cos(\frac{11\pi}{6})=\cos(2\pi-\frac{\pi}{6})=\cos(\frac{\pi}{6})=\frac{\sqrt{3}}{2}\)

Step3: Apply the half - angle formula

Since \(\frac{11\pi}{12}\) is in the second quadrant (\(\frac{\pi}{2}<\frac{11\pi}{12}<\pi\)) and \(\sin x>0\) in the second quadrant.
Substitute \(\alpha = \frac{11\pi}{6}\) into the half - angle formula \(\sin\frac{\alpha}{2}=\sqrt{\frac{1-\cos\alpha}{2}}\)

$$
LATEXBLOCK0
$$

We can also rationalize it further. Multiply numerator and denominator by \(\sqrt{2}\)

$$
LATEXBLOCK1
$$

Answer:

\(\frac{\sqrt{6}-\sqrt{2}}{4}\)