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