Question

which of the following best explains why \\(\cos \frac{2\pi}{3} \
eq \cos \frac{5\pi}{3}\\)?\
\\(\bigcirc\\) the angles do not have the same reference angle.\
\\(\bigcirc\\) cosine is negative in the second quadrant and positive in the fourth quadrant.\
\\(\bigcirc\\) cosine is positive in the second quadrant and negative in the fourth quadrant.\
\\(\bigcirc\\) the angles do not have the same reference angle or the same sign.

Explanation:

Step1: Find quadrants of angles

$\frac{2\pi}{3}$ is in Q2; $\frac{5\pi}{3}$ is in Q4.

Step2: Check cosine signs by quadrant

Cosine is negative in Q2, positive in Q4.

Step3: Evaluate reference angles

Ref angle of $\frac{2\pi}{3}$: $\pi-\frac{2\pi}{3}=\frac{\pi}{3}$; ref angle of $\frac{5\pi}{3}$: $2\pi-\frac{5\pi}{3}=\frac{\pi}{3}$ (same ref angle).

Step4: Eliminate wrong options

Options 1,3,4 are incorrect (same ref angle; cosine sign rule wrong in 3; 4 has wrong ref angle claim).

Answer:

B. Cosine is negative in the second quadrant and positive in the fourth quadrant.