$Find the angle that corresponds to a given point on the unit circle.$
Answer 1
Ella Lewis
Let’s consider the point (\frac{\sqrt{3}}{2}, \, \frac{1}{2}) on the unit circle. This point lies in the first quadrant and has coordinates (cos(\theta), sin(\theta)). We need to find the angle \theta that corresponds to this point.
Using the coordinates, we know that
$ \cos(\theta) = \frac{\sqrt{3}}{2} \quad \text{and} \quad \sin(\theta) = \frac{1}{2} $
The angle \theta that satisfies both these conditions is
$ \theta = \frac{\pi}{6} $
Therefore, the angle corresponding to the point (\frac{\sqrt{3}}{2}, \frac{1}{2}) is \frac{\pi}{6} radians.
Answer 2
Olivia Lee
Suppose we are given the point (frac{-1}{2}, , frac{sqrt{3}}{2}) on the unit circle. This point is located in the second quadrant with coordinates (cos( heta), sin( heta)). We need to determine the angle heta.
According to the coordinates, we have
$ cos( heta) = frac{-1}{2} quad ext{and} quad sin( heta) = frac{sqrt{3}}{2} $
The angle heta that satisfies these conditions is
$ heta = frac{2pi}{3} $
Hence, the angle corresponding to the point (frac{-1}{2}, frac{sqrt{3}}{2}) is frac{2pi}{3} radians.
Answer 3
Ava Martin
Consider the point (frac{-sqrt{2}}{2}, , frac{sqrt{2}}{2}). This point falls in the second quadrant.
The coordinates give us
$ cos( heta) = frac{-sqrt{2}}{2} quad ext{and} quad sin( heta) = frac{sqrt{2}}{2} $
The angle heta is
$ heta = frac{3pi}{4} $
Thus, heta for the point (frac{-sqrt{2}}{2}, frac{sqrt{2}}{2}) is frac{3pi}{4} radians.
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