Question

Evaluate the expression without using a calculator. $$ \arccos \frac{1}{2} $$

Short answer

\(\arccos \frac{1}{2} = 60^\circ\) or \(\frac{\pi}{3}\) in radians

Step-by-step solution

Familiarize with the Unit Circle

The unit circle is a circle of unit radius, centered at the origin of the Cartesian plane. The x-coordinate of a point on the unit circle represents a cosine of the angle, while the y-coordinate represents the sine. If we can find an angle which has cosine value equal to \(\frac{1}{2}\), this means that the arccosine of \(\frac{1}{2}\) will be the angle.

Identify the Angle with Cosine equals to \(\frac{1}{2}\)

Looking at the unit circle and recognize that cosine is positive in the first and fourth quadrants. However, the range of the inverse cosine function is only from 0 to \(\pi\) (or from 0 to 180 degrees), so we rule out the fourth quadrant. Thus we need an angle in the first quadrant. We recall that \(\cos(60^\circ) = \frac{1}{2}\) or \(\cos(\frac{\pi}{3}) = \frac{1}{2}\), so \(\arccos \frac{1}{2}\) will be equals to \(60^\circ\) or \(\frac{\pi}{3}\) in radians.