evaluate sin (23π/6)

Explanation:

Step1: Simplify the angle

$$\sin\left(\frac{23\pi}{6}\right)=\sin\left(4\pi-\frac{\pi}{6}\right)$$

Step2: Use the periodicity and induction formula

Since $\sin(x + 2k\pi)=\sin x$ ($k\in\mathbb{Z}$), and $\sin(2\pi - \alpha)=-\sin\alpha$. Here $k = 2$, $x=-\frac{\pi}{6}$, so $\sin\left(4\pi-\frac{\pi}{6}\right)=\sin\left(-\frac{\pi}{6}\right)$ And $\sin\left(-\frac{\pi}{6}\right)=-\sin\frac{\pi}{6}$

Step3: Calculate the value

Since $\sin\frac{\pi}{6}=\frac{1}{2}$, then $-\sin\frac{\pi}{6}=-\frac{1}{2}$

Answer:

$-\frac{1}{2}$