omar wants to determine cos(-\\frac{2\\pi}{3}) using the ratios of the 30° - 60° - 90° right triangle. how…

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

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

Answer

Explanation:

Step1: Convert angle to degrees

First, convert $-\frac{2\pi}{3}$ to degrees. We know that $\pi$ radians is $180^{\circ}$, so $-\frac{2\pi}{3}=-\frac{2}{3}\times180^{\circ}=- 120^{\circ}$.

Step2: Find the reference - angle

The reference - angle of an angle $\theta$ in the third quadrant is $\theta + 180^{\circ}$. For $\theta=-120^{\circ}$, the reference - angle is $180^{\circ}-120^{\circ} = 60^{\circ}$.

Step3: Determine the cosine value

In a $30 - 60-90$ right - triangle, $\cos60^{\circ}=\frac{1}{2}$. In the third quadrant, cosine is negative. So, $\cos(-\frac{2\pi}{3})=-\frac{1}{2}$.

Answer:

Use the reference angle of $60^{\circ}$ and the fact that $-\frac{2\pi}{3}$ is in the third quadrant to determine $\cos(-\frac{2\pi}{3})=-\frac{1}{2}$.