Find the cosine of an angle using the unit circle in the complex plane
Answer 1
Henry Green
Given an angle \( \theta \) in the complex plane, the unit circle can be used to find the cosine of the angle. The cosine of the angle \( \theta \) is the x-coordinate of the point where the terminal side of the angle intersects the unit circle.
Let’s consider \( \theta = \frac{\pi}{4} \), find \( \cos(\theta) \).
On the unit circle, the coordinates of the point at angle \( \frac{\pi}{4} \) are \( \left( \cos \left( \frac{\pi}{4} \right), \sin \left( \frac{\pi}{4} \right) \right) \).
Since \( \frac{\pi}{4} \) is a 45-degree angle, the coordinates are \( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \).
Thus, the cosine of \( \frac{\pi}{4} \) is:
$ \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} $
Answer 2
Mia Harris
Given an angle ( heta ) in the complex plane, the unit circle can be used to find the cosine. The cosine of the angle ( heta ) is the x-coordinate of the point where the angle intersects the unit circle.
Consider ( heta = 0 ). The point on the unit circle at ( heta = 0 ) is ( (1, 0) ).
Thus, the cosine of ( heta = 0 ) is:
$ cos(0) = 1 $
Answer 3
Alex Thompson
Find the cosine of the angle ( frac{pi}{2} ) using the unit circle in the complex plane.
At ( heta = frac{pi}{2} ), the coordinates on the unit circle are ( (0, 1) ).
Therefore, the cosine of ( frac{pi}{2} ) is:
$ cos left( frac{pi}{2}
ight) = 0 $
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