Question

Find the exact value of the given expression. $$ \cos ^{-1} 0 $$

Short answer

The exact value of the given expression is \(\cos^{-1}(0) = \frac{π}{2}\).

Step-by-step solution

Recall the definition of the inverse cosine function

The inverse cosine function, \(\cos ^{-1}\), is defined as the angle θ for which the cosine of θ is equal to a given value. In other words, if \(\cos(\theta) = x\), then \(\cos^{-1}(x) = \theta\). We are asked to find the angle \(\theta\) when the cosine of θ is 0, that is, we want to find \(\cos^{-1}(0)\).

Analyze the unit circle

We can use the unit circle to find the angle θ for which the cosine is equal to 0. The cosine function on the unit circle corresponds to the x-coordinate of a point on the circle. We want to find an angle θ such that the x-coordinate on the unit circle is 0. This occurs on the circle at two points: (0,1) and (0,-1). The coordinates correspond to the angles \(θ =\frac{π}{2}\) and \(θ =\frac{3π}{2}\), but since the inverse cosine function returns angles in the range of \(0 \leq θ \leq π\), we choose the angle \(θ =\frac{π}{2}\).

Write the final answer

Therefore, the exact value of the given expression is \[ \cos ^{-1} (0) = \frac{π}{2} \]