Find the sine and cosine values for an angle of $frac{5pi}{4}$ radians on the unit circle.
Answer 1
Ava Martin
To find the sine and cosine values for an angle of $\frac{5\pi}{4}$ radians, we need to locate this angle on the unit circle.
First, we recognize that $\frac{5\pi}{4}$ radians is in the third quadrant because it is more than $\pi$ radians (180 degrees) but less than $\frac{3\pi}{2}$ radians (270 degrees).
The reference angle for $\frac{5\pi}{4}$ radians is $\frac{5\pi}{4} – \pi = \frac{\pi}{4}$ radians.
In the third quadrant, the sine and cosine values are both negative. For the reference angle $\frac{\pi}{4}$, the sine and cosine values are both $\frac{\sqrt{2}}{2}$.
Therefore, the sine and cosine values for $\frac{5\pi}{4}$ radians are:
$\sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$
$\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$
Answer 2
Joseph Robinson
To determine the sine and cosine values for $frac{5pi}{4}$ radians, we use the unit circle.
$frac{5pi}{4}$ radians is located in the third quadrant because it is greater than $pi$ (180 degrees) but less than $2pi$ (360 degrees).
The reference angle for $frac{5pi}{4}$ is $frac{pi}{4}$.
For $frac{pi}{4}$, the sine and cosine values are both $frac{sqrt{2}}{2}$.
Since the angle is in the third quadrant, both sine and cosine values will be negative, giving:
$sinleft(frac{5pi}{4}
ight) = -frac{sqrt{2}}{2}$
$cosleft(frac{5pi}{4}
ight) = -frac{sqrt{2}}{2}$
Answer 3
Samuel Scott
To find the sine and cosine for $frac{5pi}{4}$ radians:
1. $frac{5pi}{4}$ is in the third quadrant.
2. Reference angle: $frac{pi}{4}$.
3. Values for $frac{pi}{4}$: $frac{sqrt{2}}{2}$.
4. In the third quadrant, both values are negative:
$sinleft(frac{5pi}{4}
ight) = -frac{sqrt{2}}{2}$
$cosleft(frac{5pi}{4}
ight) = -frac{sqrt{2}}{2}$
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