find the exact value of \\( \\sec \\frac { 5 \\pi } { 3 } \\) in simplest form with a rational denominator.

Answer:

(2)

Explanation:

Step1: Recall the definition of secant

(\sec\theta=\frac{1}{\cos\theta}), so (\sec\frac{5\pi}{3}=\frac{1}{\cos\frac{5\pi}{3}})

Step2: Find the value of (\cos\frac{5\pi}{3})

Using the unit - circle, (\cos\frac{5\pi}{3}=\cos(2\pi - \frac{\pi}{3})) Since (\cos(A - B)=\cos A\cos B+\sin A\sin B) and for (A = 2\pi,B=\frac{\pi}{3}), (\cos(2\pi)=1,\sin(2\pi)=0) (\cos(2\pi-\frac{\pi}{3})=\cos2\pi\cos\frac{\pi}{3}+\sin2\pi\sin\frac{\pi}{3}=1\times\frac{1}{2}+0\times\frac{\sqrt{3}}{2}=\frac{1}{2})

Step3: Calculate (\sec\frac{5\pi}{3})

Since (\sec\frac{5\pi}{3}=\frac{1}{\cos\frac{5\pi}{3}}) and (\cos\frac{5\pi}{3}=\frac{1}{2}) (\sec\frac{5\pi}{3}=\frac{1}{\frac{1}{2}} = 2)