41. -/2 points details my notes larpcalclimaga8 5.4.070. 0/6 submissions used find the solution(s) of the…

Explanation:

Step1: Use sum - and - difference formulas for cosine

The sum - of - angles formula for cosine is $\cos(A + B)=\cos A\cos B-\sin A\sin B$ and the difference - of - angles formula is $\cos(A - B)=\cos A\cos B+\sin A\sin B$. For $4\cos(x+\frac{7\pi}{6})-4\cos(x - \frac{7\pi}{6})=-4$, we have: [ \begin{align*} 4\left(\cos x\cos\frac{7\pi}{6}-\sin x\sin\frac{7\pi}{6}\right)-4\left(\cos x\cos\frac{7\pi}{6}+\sin x\sin\frac{7\pi}{6}\right)&=-4\ 4\cos x\cos\frac{7\pi}{6}-4\sin x\sin\frac{7\pi}{6}-4\cos x\cos\frac{7\pi}{6}-4\sin x\sin\frac{7\pi}{6}&=-4\

• 8\sin x\sin\frac{7\pi}{6}&=-4 \end{align*} ] Since $\sin\frac{7\pi}{6}=-\frac{1}{2}$, the equation becomes $-8\sin x\times(-\frac{1}{2})=-4$, which simplifies to $4\sin x=-4$, so $\sin x = - 1$.

Step2: Find solutions in the given interval

We know that $\sin x=-1$ when $x=\frac{3\pi}{2}$ in the interval $[0,2\pi)$.

Answer:

$\frac{3\pi}{2}$