what are the solutions to the equation sin(x + 7π/2)= -√3/2 over the interval 0, 2π?\nπ/6 only\nπ/6 and 5π/6…

Explanation:

Step1: Simplify the sine - function

Use the trigonometric identity $\sin(A + B)=\sin A\cos B+\cos A\sin B$. Here, $\sin(x+\frac{7\pi}{2})=\sin x\cos\frac{7\pi}{2}+\cos x\sin\frac{7\pi}{2}$. Since $\cos\frac{7\pi}{2} = 0$ and $\sin\frac{7\pi}{2}=- 1$, then $\sin(x+\frac{7\pi}{2})=-\cos x$. So the equation $\sin(x + \frac{7\pi}{2})=-\frac{\sqrt{3}}{2}$ becomes $-\cos x=-\frac{\sqrt{3}}{2}$, or $\cos x=\frac{\sqrt{3}}{2}$.

Step2: Find the solutions in the given interval

We know that $\cos x=\frac{\sqrt{3}}{2}$ when $x = 2k\pi\pm\frac{\pi}{6},k\in\mathbb{Z}$. For the interval $[0,2\pi]$, when $k = 0$, $x=\frac{\pi}{6}$; when $k = 1$, $x = 2\pi-\frac{\pi}{6}=\frac{11\pi}{6}$.

Answer:

C. $\frac{\pi}{6}$ and $\frac{11\pi}{6}$ only