Questions: Find the following in radians without a calculator. [ cos ^-1left(-fracsqrt22right)= ]

Solution Steps

To find the value of \(\cos^{-1}\left(-\frac{\sqrt{2}}{2}\right)\) in radians, we need to determine the angle whose cosine is \(-\frac{\sqrt{2}}{2}\). We know that \(\cos(\theta) = -\frac{\sqrt{2}}{2}\) at specific angles in the unit circle.

1. Identify the reference angle where \(\cos(\theta) = \frac{\sqrt{2}}{2}\).
2. Determine the angles in the unit circle where the cosine value is negative.
3. Return the principal value of the inverse cosine function.

The reference angle where \(\cos(\theta) = \frac{\sqrt{2}}{2}\) is given by: \[ \theta = \frac{\pi}{4} \]

The angles in the unit circle where \(\cos(\theta) = -\frac{\sqrt{2}}{2}\) are: \[ \theta_1 = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \] \[ \theta_2 = \pi + \frac{\pi}{4} = \frac{5\pi}{4} \]

The principal value of the inverse cosine function, which is the angle in the range \([0, \pi]\), is: \[ \cos^{-1}\left(-\frac{\sqrt{2}}{2}\right) = \frac{3\pi}{4} \]

Final Answer