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 = 8π/3. sec 8π/3 = □ (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Rewrite the angle

First, rewrite $\frac{8\pi}{3}$ as $\frac{6\pi + 2\pi}{3}=2\pi+\frac{2\pi}{3}$. Since adding $2\pi$ to an angle does not change the value of circular - functions, $\sec(\frac{8\pi}{3})=\sec(\frac{2\pi}{3})$.

Step2: Recall the definition of secant

Recall that $\sec\theta=\frac{1}{\cos\theta}$. So, $\sec(\frac{2\pi}{3})=\frac{1}{\cos(\frac{2\pi}{3})}$.

Step3: Find the cosine value on the unit - circle

On the unit - circle, for $\theta = \frac{2\pi}{3}$, the $x$ - coordinate (which is $\cos\theta$) is $-\frac{1}{2}$. So, $\cos(\frac{2\pi}{3})=-\frac{1}{2}$.

Step4: Calculate the secant value

Substitute $\cos(\frac{2\pi}{3}) = -\frac{1}{2}$ into the secant formula: $\sec(\frac{2\pi}{3})=\frac{1}{\cos(\frac{2\pi}{3})}=\frac{1}{-\frac{1}{2}}=-2$.

Answer:

$-2$