Find the exact values of the trigonometric functions for the angle $ frac{5pi}{3} $ in the unit circle
Answer 1
Sophia Williams
To find the exact values of the trigonometric functions for the angle $ \frac{5\pi}{3} $ in the unit circle, we first note that:
$ \frac{5\pi}{3} = 2\pi – \frac{\pi}{3} $
This means the angle is located in the fourth quadrant.
We can use the reference angle of $ \frac{\pi}{3} $:
$ \cos \left( \frac{5\pi}{3} \right) = \cos \left( 2\pi – \frac{\pi}{3} \right) = \cos \left( \frac{\pi}{3} \right) = \frac{1}{2} $
$ \sin \left( \frac{5\pi}{3} \right) = \sin \left( 2\pi – \frac{\pi}{3} \right) = -\sin \left( \frac{\pi}{3} \right) = -\frac{\sqrt{3}}{2} $
$ \tan \left( \frac{5\pi}{3} \right) = \frac{\sin \left( \frac{5\pi}{3} \right)}{\cos \left( \frac{5\pi}{3} \right)} = \frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}} = -\sqrt{3} $
Answer 2
Emily Hall
We know $ frac{5pi}{3} = 2pi – frac{pi}{3} $, which places the angle in the fourth quadrant.
The reference angle is $ frac{pi}{3} $:
$ cos left( frac{5pi}{3}
ight) = cos left( frac{pi}{3}
ight) = frac{1}{2} $
$ sin left( frac{5pi}{3}
ight) = -sin left( frac{pi}{3}
ight) = -frac{sqrt{3}}{2} $
$ an left( frac{5pi}{3}
ight) = -sqrt{3} $
Answer 3
James Taylor
We know $ frac{5pi}{3} $ is in the fourth quadrant with a reference angle of $ frac{pi}{3} $.
Thus,
$ cos left( frac{5pi}{3}
ight) = frac{1}{2} $
$ sin left( frac{5pi}{3}
ight) = -frac{sqrt{3}}{2} $
$ an left( frac{5pi}{3}
ight) = -sqrt{3} $
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