which expression is equivalent to sin(7π/6)?\no sin(π/6)\no sin(5π/6)\no sin(5π/3)\no sin(11π/6)

Explanation:

Step1: Rewrite $\sin\frac{7\pi}{6}$

We know that $\frac{7\pi}{6}=\pi+\frac{\pi}{6}$. So, $\sin\frac{7\pi}{6}=\sin(\pi + \frac{\pi}{6})$.

Step2: Apply the trig - identity $\sin(A + B)$

The identity $\sin(A + B)=\sin A\cos B+\cos A\sin B$. When $A = \pi$ and $B=\frac{\pi}{6}$, $\sin(\pi+\frac{\pi}{6})=\sin\pi\cos\frac{\pi}{6}+\cos\pi\sin\frac{\pi}{6}$. Since $\sin\pi = 0$ and $\cos\pi=- 1$, we have $\sin(\pi+\frac{\pi}{6})=0\times\cos\frac{\pi}{6}+(-1)\times\sin\frac{\pi}{6}=-\sin\frac{\pi}{6}$.

Step3: Analyze the periodicity of the sine function

The sine function has a period of $2\pi$, i.e., $\sin(x + 2k\pi)=\sin x,k\in\mathbb{Z}$. Also, $\sin(2\pi - x)=-\sin x$. We know that $\sin\frac{11\pi}{6}=\sin(2\pi-\frac{\pi}{6})$. Using the identity $\sin(2\pi - x)=-\sin x$, we get $\sin\frac{11\pi}{6}=-\sin\frac{\pi}{6}$. And since $\sin\frac{7\pi}{6}=-\sin\frac{\pi}{6}$, then $\sin\frac{7\pi}{6}=\sin\frac{11\pi}{6}$.

Answer:

$\sin\frac{11\pi}{6}$