which expression is equivalent to $sin\frac{7pi}{6}$?\n$sin\frac{pi}{6}$\n$sin\frac{5pi}{6}$\n$sin\frac{5pi}{…

which expression is equivalent to $sin\frac{7pi}{6}$?\n$sin\frac{pi}{6}$\n$sin\frac{5pi}{6}$\n$sin\frac{5pi}{3}$\n$sin\frac{11pi}{6}$

which expression is equivalent to $sin\frac{7pi}{6}$?\n$sin\frac{pi}{6}$\n$sin\frac{5pi}{6}$\n$sin\frac{5pi}{3}$\n$sin\frac{11pi}{6}$

Answer

Explanation:

Step1: Analyze the angle $\frac{7\pi}{6}$

We know that $\frac{7\pi}{6}=\pi+\frac{\pi}{6}$. According to the trigonometric identity $\sin(A + \pi)=-\sin A$, so $\sin\frac{7\pi}{6}=\sin(\pi+\frac{\pi}{6})=-\sin\frac{\pi}{6}$.

Step2: Analyze each option

  • $\sin\frac{\pi}{6}\neq-\sin\frac{\pi}{6}$.
  • $\sin\frac{5\pi}{6}=\sin(\pi - \frac{\pi}{6})=\sin\frac{\pi}{6}\neq-\sin\frac{\pi}{6}$.
  • $\sin\frac{5\pi}{3}=\sin(2\pi-\frac{\pi}{3})=-\sin\frac{\pi}{3}\neq-\sin\frac{\pi}{6}$.
  • $\sin\frac{11\pi}{6}=\sin(2\pi-\frac{\pi}{6})=-\sin\frac{\pi}{6}$.

Answer:

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