use the unit circle, along with the definitions of the circular functions, to find the exact value for the…

Explanation:

Step1: Rewrite the angle

First, rewrite $\frac{11\pi}{3}$ as an equivalent angle between $0$ and $2\pi$. We know that $\frac{11\pi}{3}=4\pi-\frac{\pi}{3}$. Since the trigonometric - functions are periodic with period $2\pi$, $\sec(\frac{11\pi}{3})=\sec(4\pi - \frac{\pi}{3})=\sec(-\frac{\pi}{3})$. And since $\sec(-\theta)=\sec(\theta)$ (because $\sec\theta=\frac{1}{\cos\theta}$ and $\cos(-\theta)=\cos\theta$), then $\sec(\frac{11\pi}{3})=\sec(\frac{\pi}{3})$.

Step2: Recall the definition of secant

Recall that $\sec\theta=\frac{1}{\cos\theta}$. For $\theta = \frac{\pi}{3}$, on the unit - circle, $\cos\frac{\pi}{3}=\frac{1}{2}$.

Step3: Calculate the secant value

Using the formula $\sec\theta=\frac{1}{\cos\theta}$, when $\theta=\frac{\pi}{3}$, we have $\sec\frac{\pi}{3}=\frac{1}{\cos\frac{\pi}{3}}$. Substituting $\cos\frac{\pi}{3}=\frac{1}{2}$ into the formula, we get $\sec\frac{\pi}{3}=2$.

Answer:

$2$