which expression is equivalent to sin(7π/6)? o sin(π/6) o sin(5π/6) o sin(5π/3) o sin(11π/6)

Explanation:

Step1: Use the sine - angle formula

The sine function has the property $\sin(x)=\sin(2\pi - x)$. Also, $\sin(x)=\sin(\pi + (x - \pi))=-\sin(x - \pi)$. We know that $\sin\left(\frac{7\pi}{6}\right)=\sin\left(\pi+\frac{\pi}{6}\right)$. By the formula $\sin(A + B)=\sin A\cos B+\cos A\sin B$, when $A = \pi$ and $B=\frac{\pi}{6}$, we have $\sin\left(\pi+\frac{\pi}{6}\right)=\sin\pi\cos\frac{\pi}{6}+\cos\pi\sin\frac{\pi}{6}=0\times\frac{\sqrt{3}}{2}+(- 1)\times\frac{1}{2}=-\frac{1}{2}$.

Step2: Analyze each option

• $\sin\left(\frac{\pi}{6}\right)=\frac{1}{2}$
• $\sin\left(\frac{5\pi}{6}\right)=\sin\left(\pi-\frac{\pi}{6}\right)=\sin\frac{\pi}{6}=\frac{1}{2}$
• $\sin\left(\frac{5\pi}{3}\right)=\sin\left(2\pi-\frac{\pi}{3}\right)=-\sin\frac{\pi}{3}=-\frac{\sqrt{3}}{2}$
• $\sin\left(\frac{11\pi}{6}\right)=\sin\left(2\pi-\frac{\pi}{6}\right)=-\sin\frac{\pi}{6}=-\frac{1}{2}$

Answer:

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