Question

Evaluate the expression without using a calculator. $$ \arccos 0 $$

Short answer

\( \arccos 0 = \frac{\pi}{2} \)

Step-by-step solution

Understanding the arccosine function

The arccosine function, represented as \( \arccos \), is the inverse of the cosine function. It undoes what the cosine function does. If the cosine of an angle is known, the arccosine function can be used to find the angle. Here, we need to find the angle whose cosine equality equals 0.

Applying the definition on the unit circle

In the context of the unit circle, the cosine of an angle represents the x-coordinate of the point where the terminal side of the angle intersects the unit circle. Knowing this, we are searching for the angle whose terminal side intersects the x-axis at 0. The angles on the unit circle that satisfy this condition are \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \). However, the arccosine function only provides values in the interval [0,π], it 'forgets' about anything more than one complete turn about the origin which rules out \( \frac{3\pi}{2} \).

Final Answer

So, the angle that has a cosine value of 0 is \( \frac{\pi}{2} \) or 90 degrees. Hence, \( \arccos 0 = \frac{\pi}{2} \).