Questions: Evaluate the following expressions. Your answer must be an exact angle in radians and in the interval [ 0, π ] . Example: Enter pi/6 for π/6. (a) cos^(-1)(sqrt(2)/2)= (b) cos^(-1)(0)= (c) cos^(-1)(-1)=

Solution Steps

The inverse cosine function \(\cos^{-1}(x)\) returns the angle \(\theta\) in the interval \([0, \pi]\) such that \(\cos(\theta) = x\).

We know that: \[ \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \] Thus: \[ \cos^{-1}\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4} \]

We need to find the angle \(\theta\) in the interval \([0, \pi]\) such that \(\cos(\theta) = 0\).

We know that: \[ \cos\left(\frac{\pi}{2}\right) = 0 \] Thus: \[ \cos^{-1}(0) = \frac{\pi}{2} \]

We need to find the angle \(\theta\) in the interval \([0, \pi]\) such that \(\cos(\theta) = -1\).

We know that: \[ \cos(\pi) = -1 \] Thus: \[ \cos^{-1}(-1) = \pi \]

Final Answer