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 reference angles

For \(\theta = \frac{2\pi}{3}\), it's in the second quadrant. Reference angle: \(\pi - \frac{2\pi}{3}=\frac{\pi}{3}\). Cosine is negative here.
For \(\theta = \frac{5\pi}{3}\), it's in the fourth quadrant. Reference angle: \(2\pi - \frac{5\pi}{3}=\frac{\pi}{3}\). Cosine is positive here.

Step2: Analyze options

• Option 1: Both have reference angle \(\frac{\pi}{3}\), so wrong.
• Option 2: Cosine is negative in second ( \(\frac{2\pi}{3}\) ) and positive in fourth ( \(\frac{5\pi}{3}\) ). This explains why their cosines differ (sign difference, even same reference angle).
• Option 3: Cosine is negative in second, positive in fourth (opposite of this option), wrong.
• Option 4: They have the same reference angle (\(\frac{\pi}{3}\)), wrong.

Answer:

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