Questions: cos^(-1)(-(sqrt(3)/2))

Solution Steps

We are tasked with finding $\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right)$. The value $-\frac{\sqrt{3}}{2}$ indicates that we are looking for an angle whose cosine is negative.

The reference angle corresponding to $\frac{\sqrt{3}}{2}$ is calculated as: $$\text{reference angle} = \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}$$

The cosine function is negative in the second and third quadrants. Therefore, we need to find the angles in these quadrants that correspond to the reference angle.

1. Second Quadrant: $$\text{angle}_{\text{second quadrant}} = \pi - \text{reference angle} = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$$

2. Third Quadrant: $$\text{angle}_{\text{third quadrant}} = \pi + \text{reference angle} = \pi + \frac{\pi}{3} = \frac{4\pi}{3}$$

Since $\cos^{-1}$ returns the angle in the range $[0, \pi]$, we select the angle from the second quadrant: $$\text{result angle} = \frac{2\pi}{3}$$

To express the result in degrees: $$\text{result angle in degrees} = \frac{2\pi}{3} \times \frac{180}{\pi} = 120$$

Final Answer