$ ext{Find the sine and cosine of an angle given in radians on the unit circle.}$
Answer 1
Michael Moore
Given an angle \( \theta = \frac{\pi}{4} \), find the sine and cosine of the angle on the unit circle.
Using the unit circle, the coordinates of the point at \( \theta = \frac{\pi}{4} \) are given by: \( (\cos(\frac{\pi}{4}), \sin(\frac{\pi}{4})) \).
We know that:
$ \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} $
$ \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} $
Therefore, the sine and cosine of the angle \( \theta = \frac{\pi}{4} \) are both \( \frac{\sqrt{2}}{2} \).
Answer 2
Christopher Garcia
Consider the angle ( heta = frac{pi}{3} ). Find the sine and cosine of ( heta ) on the unit circle.
For ( heta = frac{pi}{3} ), the coordinates of the point on the unit circle are ( (cos(frac{pi}{3}), sin(frac{pi}{3})) ).
We find:
$ cos(frac{pi}{3}) = frac{1}{2} $
$ sin(frac{pi}{3}) = frac{sqrt{3}}{2} $
Thus, the cosine of the angle ( heta = frac{pi}{3} ) is ( frac{1}{2} ) and the sine is ( frac{sqrt{3}}{2} ).
Answer 3
Chloe Evans
Determine the sine and cosine of ( heta = frac{pi}{6} ) on the unit circle.
At ( heta = frac{pi}{6} ), the coordinates are ( (cos(frac{pi}{6}), sin(frac{pi}{6})) ).
So, we have:
$ cos(frac{pi}{6}) = frac{sqrt{3}}{2} $
$ sin(frac{pi}{6}) = frac{1}{2} $
Therefore, the cosine is ( frac{sqrt{3}}{2} ) and the sine is ( frac{1}{2} ).
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