Find the sine and cosine of the angle $ frac{5pi}{4} $ on the unit circle.
Answer 1
Isabella Walker
To find the sine and cosine of the angle \( \frac{5\pi}{4} \) on the unit circle, we start by locating the angle.
The angle \( \frac{5\pi}{4} \) is in the third quadrant of the unit circle.
We recognize that \( \frac{5\pi}{4} \) is the same as \( \pi + \frac{\pi}{4} \).
In the third quadrant, both sine and cosine are negative.
Now, take the reference angle \( \frac{\pi}{4} \), which has sine and cosine values of \( \frac{1}{\sqrt{2}} \) or \( \frac{\sqrt{2}}{2} \).
Since we are in the third quadrant, we apply the negative signs:
$ \sin \left( \frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2} $
$ \cos \left( \frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2} $
Answer 2
Joseph Robinson
To determine the sine and cosine of the angle ( frac{5pi}{4} ) on the unit circle, let’s first locate the angle.
The angle ( frac{5pi}{4} ) is in the third quadrant of the unit circle.
Since ( frac{5pi}{4} = pi + frac{pi}{4} ), we use the reference angle ( frac{pi}{4} ).
The sine and cosine of ( frac{pi}{4} ) are both ( frac{sqrt{2}}{2} ).
In the third quadrant, both sine and cosine are negative:
$ sin left( frac{5pi}{4}
ight) = -frac{sqrt{2}}{2} $
$ cos left( frac{5pi}{4}
ight) = -frac{sqrt{2}}{2} $
Answer 3
James Taylor
To find the sine and cosine of ( frac{5pi}{4} ), note the angle is in the third quadrant.
Use the reference angle ( frac{pi}{4} ), which gives:
$ sin left( frac{5pi}{4}
ight) = -frac{sqrt{2}}{2} $
$ cos left( frac{5pi}{4}
ight) = -frac{sqrt{2}}{2} $
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