arccos(1/2) in terms of pi
Understand the Problem
The question is asking for the value of the inverse cosine function (arccos) of 1/2 expressed in terms of pi.
Answer
The value of $\text{arccos}\left(\frac{1}{2}\right)$ is $\frac{\pi}{3}$.
Answer for screen readers
The value of $\text{arccos}\left(\frac{1}{2}\right)$ is $\frac{\pi}{3}$.
Steps to Solve
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Identify the function
We need to find the value of the inverse cosine function, which is denoted as $\cos^{-1}(x)$ or $\text{arccos}(x)$. In this case, we are looking for $\text{arccos}\left(\frac{1}{2}\right)$. -
Evaluate inverse cosine
The inverse cosine function gives us an angle whose cosine is the given value. We need to determine the angle $\theta$ such that
$$ \cos(\theta) = \frac{1}{2} $$ -
Determine the angle
We know that the cosine of $\frac{\pi}{3}$ (or 60 degrees) is $\frac{1}{2}$. Therefore, we have:
$$ \theta = \frac{\pi}{3} $$ -
Express the result
Thus, we can conclude that
$$ \text{arccos}\left(\frac{1}{2}\right) = \frac{\pi}{3} $$
The value of $\text{arccos}\left(\frac{1}{2}\right)$ is $\frac{\pi}{3}$.
More Information
The inverse cosine function is widely used in trigonometry and plays an important role in determining angles from cosine values. The value $\frac{\pi}{3}$ is a standard angle in trigonometric calculations, corresponding to 60 degrees.
Tips
- A common mistake is confusing the inverse cosine with the cosine function itself, leading to incorrect angle estimations. Always remember to look for the angle that corresponds to the given cosine value.
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