Question

use the unit - circle, along with the definitions of the circular functions, to find the exact value for the function at the right, given s = 7π/3. sec(7π/3)= (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall secant definition

$\sec(x)=\frac{1}{\cos(x)}$, so $\sec(\frac{7\pi}{3})=\frac{1}{\cos(\frac{7\pi}{3})}$.

Step2: Rewrite the angle

$\frac{7\pi}{3}=2\pi+\frac{\pi}{3}$. Since $\cos(x + 2k\pi)=\cos(x)$ for any real - number $x$ and integer $k$, then $\cos(\frac{7\pi}{3})=\cos(2\pi+\frac{\pi}{3})=\cos(\frac{\pi}{3})$.

Step3: Find cosine value

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

Step4: Calculate secant value

Since $\sec(\frac{7\pi}{3})=\frac{1}{\cos(\frac{7\pi}{3})}$ and $\cos(\frac{7\pi}{3})=\frac{1}{2}$, then $\sec(\frac{7\pi}{3}) = 2$.

Answer:

$2$