Find the angle in radians corresponding to a point on the unit circle given coordinates $(x, y)$

To find the angle θ corresponding to the point $(\frac{1}{2}, \frac{\sqrt{3}}{2})$ on the unit circle, we start with the basic trigonometric relationships:

$ \cos \theta = x $

$ \sin \theta = y $

Given $ x = \frac{1}{2} $ and $ y = \frac{\sqrt{3}}{2} $, we can use the inverse trigonometric functions:

$ \theta = \cos^{-1}\left(\frac{1}{2}\right) $

$ \theta = \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) $

We know that:

$ \cos \left(\frac{\pi}{3}\right) = \frac{1}{2} $

$ \sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} $

Therefore, the angle corresponding to the given point is:

$ \theta = \frac{\pi}{3} $

To determine the angle θ for the point $(frac{1}{2}, frac{sqrt{3}}{2})$ on the unit circle, use the trigonometric identities:

$ cos heta = frac{1}{2} $

$ sin heta = frac{sqrt{3}}{2} $

Knowing the standard values, angle θ can be calculated as:

$ heta = arccos left( frac{1}{2}
ight) $

The reference angle for these values is commonly known:

$ cos left( frac{pi}{3}
ight) = frac{1}{2} $

Therefore:

$ heta = frac{pi}{3} $

Thus, the angle in radians is:

$ heta = frac{pi}{3} $

The coordinates $(frac{1}{2}, frac{sqrt{3}}{2})$ on the unit circle correspond to the angle:

$ cos heta = frac{1}{2} $

The standard angle with this value is:

$ heta = frac{pi}{3} $

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