find the exact value of the given expression without using a calculator. cos(-13π/3) cos(-13π/3)=□ (simplify…

Explanation:

Step1: Use cosine - angle property

Since $\cos(-\alpha)=\cos(\alpha)$, then $\cos\left(-\frac{13\pi}{3}\right)=\cos\left(\frac{13\pi}{3}\right)$.

Step2: Rewrite the angle

We know that $2\pi$ is one full - rotation. We can rewrite $\frac{13\pi}{3}$ as $\frac{12\pi + \pi}{3}=4\pi+\frac{\pi}{3}$. Since $\cos(x + 2k\pi)=\cos(x)$ for any real number $x$ and integer $k$, and here $x = \frac{\pi}{3}$, $k = 2$, then $\cos\left(4\pi+\frac{\pi}{3}\right)=\cos\left(\frac{\pi}{3}\right)$.

Step3: Recall the cosine value

We know that $\cos\left(\frac{\pi}{3}\right)=\frac{1}{2}$.

Answer:

$\frac{1}{2}$