Question

evaluate. write your answer in simplified, rationalized form. do not round.
\\( \sin \left( \frac { 1 1 \pi } { 6 } \
ight) = \\)

Explanation:

Step1: Use the angle - reduction formula

We know that \(\sin(x)=\sin(2\pi - x)\) when \(x\) is in the fourth - quadrant. Here, \(\frac{11\pi}{6}=2\pi-\frac{\pi}{6}\). So, \(\sin(\frac{11\pi}{6})=\sin(2\pi - \frac{\pi}{6})\).
According to the formula \(\sin(A - B)=\sin A\cos B-\cos A\sin B\), when \(A = 2\pi\) and \(B=\frac{\pi}{6}\), since \(\sin(2\pi)=0\) and \(\cos(2\pi)=1\), we have \(\sin(2\pi-\frac{\pi}{6})=\sin(2\pi)\cos(\frac{\pi}{6})-\cos(2\pi)\sin(\frac{\pi}{6})\).
Substituting the values, we get \(\sin(2\pi-\frac{\pi}{6})=0\times\frac{\sqrt{3}}{2}-1\times\frac{1}{2}\).
Another way: Using the property \(\sin(2k\pi - \alpha)=-\sin\alpha\) (\(k\in\mathbb{Z}\)), when \(k = 1\) and \(\alpha=\frac{\pi}{6}\), \(\sin(\frac{11\pi}{6})=-\sin(\frac{\pi}{6})\).

Step2: Evaluate \(\sin(\frac{\pi}{6})\)

We know that for a right - triangle with an angle of \(\frac{\pi}{6}\) (or \(30^{\circ}\)), \(\sin(\frac{\pi}{6})=\frac{1}{2}\).

Answer:

\(-\frac{1}{2}\)