Question

omar wants to determine cos(-2π/3) using the ratios of the 30° - 60° - 90° right - triangle. how should he use this information to determine cos(-2π/3)? use the reference angle of 60° and the fact that -2π/3 is in the third quadrant to determine cos(-2π/3)= -1/2. use the reference angle of 30° and the fact that -2π/3 is in the third quadrant to determine cos(-2π/3)= -√3/2. use the reference angle of 60° and the fact that -2π/3 is in the fourth quadrant to determine cos(-2π/3)= √3/2. use the reference angle of 30° and the fact that -2π/3 is in the fourth quadrant to determine cos(-2π/3)= -1/2.

Explanation:

Step1: Recall the angle - conversion

First, note that $-\frac{2\pi}{3}$ radians is equivalent to $- 120^{\circ}$. Since $-120^{\circ}+360^{\circ}=240^{\circ}$, and $240^{\circ}$ is in the third quadrant. The reference angle of an angle $\theta$ in the third quadrant is $\theta - 180^{\circ}$. For $\theta = 240^{\circ}$, the reference angle is $240^{\circ}-180^{\circ}=60^{\circ}$.

Step2: Recall the cosine - sign in the third quadrant

In the third quadrant, the cosine function is negative. The cosine of the reference angle $60^{\circ}$ is $\cos(60^{\circ})=\frac{1}{2}$. So, $\cos(-\frac{2\pi}{3})=-\frac{1}{2}$ because of the negative sign of cosine in the third quadrant.

Answer:

He should use the reference angle of $60^{\circ}$ and the fact that $-\frac{2\pi}{3}$ is in the third quadrant to determine $\cos(-\frac{2\pi}{3})$.