evaluate the expression cos⁻¹(sin(5π/3)). give your answer as an exact value

Explanation:

Step1: Find the value of $\sin(\frac{5\pi}{3})$

We know that $\sin(\frac{5\pi}{3})=\sin(2\pi - \frac{\pi}{3})$. Using the identity $\sin(2\pi - \alpha)=-\sin\alpha$, we have $\sin(\frac{5\pi}{3})=-\sin(\frac{\pi}{3})=-\frac{\sqrt{3}}{2}$.

Step2: Evaluate $\cos^{-1}(-\frac{\sqrt{3}}{2})$

Let $\theta=\cos^{-1}(-\frac{\sqrt{3}}{2})$. Then $\cos\theta = -\frac{\sqrt{3}}{2}$, and the range of $y = \cos^{-1}x$ is $[0,\pi]$. In this range, when $\cos\theta=-\frac{\sqrt{3}}{2}$, $\theta=\frac{5\pi}{6}$.

Answer:

$\frac{5\pi}{6}$