Find the cosine and sine values of an angle in the unit circle
Answer 1
Emma Johnson
Given an angle of \( \frac{5\pi}{4} \) radians, determine the cosine and sine values using the unit circle.
First, locate the angle \( \frac{5\pi}{4} \) on the unit circle. This angle is in the third quadrant where both sine and cosine values are negative.
The reference angle for \( \frac{5\pi}{4} \) is \( \frac{\pi}{4} \). In the unit circle, the sine and cosine of \( \frac{\pi}{4} \) are both \( \frac{\sqrt{2}}{2} \).
Since \( \frac{5\pi}{4} \) is in the third quadrant, the values become negative:
$ \cos \left( \frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2} $
$ \sin \left( \frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2} $
Answer 2
James Taylor
To find the cosine and sine of the angle ( frac{5pi}{4} ), we start by noting that this angle is in the third quadrant of the unit circle.
The reference angle for ( frac{5pi}{4} ) is ( frac{pi}{4} ).
In the first quadrant, the coordinates (cosine, sine) for ( frac{pi}{4} ) are ( left( frac{sqrt{2}}{2}, frac{sqrt{2}}{2}
ight) ).
Since ( frac{5pi}{4} ) is in the third quadrant, both coordinates will be negative:
$ cos left( frac{5pi}{4}
ight) = -frac{sqrt{2}}{2} $
$ sin left( frac{5pi}{4}
ight) = -frac{sqrt{2}}{2} $
Answer 3
Isabella Walker
For the angle ( frac{5pi}{4} ) in the unit circle:
Since it is in the third quadrant, both cosine and sine values are negative.
$ cos left( frac{5pi}{4}
ight) = -frac{sqrt{2}}{2} $
$ sin left( frac{5pi}{4}
ight) = -frac{sqrt{2}}{2} $
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