Question

use the unit circle to find the value of (cos\frac{2pi}{3}) and even or odd trigonometric functions to find the value of (cosleft(-\frac{2pi}{3}
ight)). select the correct choice below and fill in any answer boxes within your choice. a. (cos\frac{2pi}{3} = square) (type an exact answer, using radicals as needed. simplify your answer. rationalize the denominator.) b. the expression is undefined. unit circle diagram with coordinates ((0,1)), ((1,0)), ((-1,0)), ((0,-1)), (left(\frac{1}{2}, \frac{sqrt{3}}{2}
ight)), (left(\frac{sqrt{3}}{2}, \frac{1}{2}
ight)), etc., and question numbers 7, 8, 9, 10, 11, 12 visible

Explanation:

Step1: Recall unit circle coordinates

On the unit circle, a point \((x,y)\) corresponds to \((\cos\theta, \sin\theta)\), where \(\theta\) is the angle. For \(\theta = \frac{2\pi}{3}\), we find the reference angle and quadrant. \(\frac{2\pi}{3}\) is in the second quadrant (between \(\frac{\pi}{2}\) and \(\pi\)), so \(\cos\theta\) is negative. The reference angle is \(\pi - \frac{2\pi}{3}=\frac{\pi}{3}\).

Step2: Find \(\cos\frac{\pi}{3}\)

We know that \(\cos\frac{\pi}{3}=\frac{1}{2}\). Since \(\frac{2\pi}{3}\) is in the second quadrant, \(\cos\frac{2\pi}{3}=-\cos\frac{\pi}{3}\).

Step3: Calculate the value

Substituting \(\cos\frac{\pi}{3}=\frac{1}{2}\) into \(-\cos\frac{\pi}{3}\), we get \(\cos\frac{2\pi}{3}=-\frac{1}{2}\).

Answer:

\(-\frac{1}{2}\)