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 = 13π/6. sec 13π/6 sec 13π/6 = □ (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Reduce the angle

Subtract \(2\pi\) from \(\frac{13\pi}{6}\) since \(2\pi=\frac{12\pi}{6}\). So \(\frac{13\pi}{6}- 2\pi=\frac{13\pi - 12\pi}{6}=\frac{\pi}{6}\). Then \(\sec\frac{13\pi}{6}=\sec\frac{\pi}{6}\).

Step2: Recall the secant - cosine relationship

Recall that \(\sec\theta=\frac{1}{\cos\theta}\). So \(\sec\frac{\pi}{6}=\frac{1}{\cos\frac{\pi}{6}}\).

Step3: Find the cosine value

On the unit - circle, \(\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}\).

Step4: Calculate the secant value

Substitute \(\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}\) into the secant formula: \(\sec\frac{\pi}{6}=\frac{1}{\frac{\sqrt{3}}{2}}=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\).

Answer:

\(\frac{2\sqrt{3}}{3}\)