use reference angles to find the exact value of the following expression. cos 5π/3 determine the reference…

Explanation:

Step1: Determine the quadrant

The angle $\frac{5\pi}{3}$ is in the fourth - quadrant since $2\pi-\frac{\pi}{3}=\frac{5\pi}{3}$ and $ \frac{3\pi}{2}<\frac{5\pi}{3}<2\pi$.

Step2: Calculate the reference angle

The reference angle $\theta'$ for an angle $\theta$ in the fourth - quadrant is given by $\theta' = 2\pi-\theta$. For $\theta=\frac{5\pi}{3}$, the reference angle $\theta'=2\pi - \frac{5\pi}{3}=\frac{6\pi - 5\pi}{3}=\frac{\pi}{3}$.

Step3: Find the cosine value

We know that $\cos\theta$ has the same absolute value as $\cos\theta'$ in the fourth - quadrant, and $\cos$ is positive in the fourth - quadrant. Since $\cos\frac{\pi}{3}=\frac{1}{2}$, then $\cos\frac{5\pi}{3}=\frac{1}{2}$.

Answer:

The reference angle is $\frac{\pi}{3}$, and $\cos\frac{5\pi}{3}=\frac{1}{2}$