arccos(1/2) in terms of pi
Understand the Problem
The question is asking for the value of the inverse cosine (arccos) of 1/2 expressed in terms of pi. This involves identifying the angle whose cosine is 1/2.
Answer
The value is $ \frac{\pi}{3} $.
Answer for screen readers
The value of $ \arccos\left(\frac{1}{2}\right) $ is $ \frac{\pi}{3} $.
Steps to Solve
- Identify the angle associated with $ \cos(x) = \frac{1}{2} $
Recall the unit circle and the values of cosine for standard angles. The cosine of an angle is equal to the x-coordinate on the unit circle.
- Determine the angles where cosine is $ \frac{1}{2} $
The values that satisfy $ \cos(x) = \frac{1}{2} $ are at the angles $ \frac{\pi}{3} $ and $ \frac{5\pi}{3} $. Since we are finding the inverse cosine, we will consider the principal value:
$$ x = \frac{\pi}{3} $$
- Specify the range of the inverse cosine function
The function $ \arccos(x) $ returns values in the range $ [0, \pi] $. Since $ \frac{\pi}{3} $ falls within this range, it is the appropriate solution.
The value of $ \arccos\left(\frac{1}{2}\right) $ is $ \frac{\pi}{3} $.
More Information
The angle $ \frac{\pi}{3} $ radians corresponds to 60 degrees. This is a standard angle in trigonometry and is commonly encountered in various geometric problems.
Tips
-
Confusing angles: Students might confuse the angles and choose $ \frac{5\pi}{3} $, which is outside the principal range of $ \arccos $. Always remember the range for $ \arccos(x) $ is $ [0, \pi] $.
-
Misinterpretation of cosine values: Not recognizing that $ \cos(x) = \frac{1}{2} $ has specific angles like $ \frac{\pi}{3} $ and $ \frac{5\pi}{3}$ can lead to incorrect conclusions.
