Find the exact values of sine and cosine for an angle of $frac{5pi}{4}$ radians on the unit circle.
Answer 1
Christopher Garcia
To solve for sine and cosine of the angle $\frac{5\pi}{4}$, we first determine its location on the unit circle.
The angle $\frac{5\pi}{4}$ radians is in the third quadrant, where both sine and cosine values are negative.
The reference angle for $\frac{5\pi}{4}$ radians is $\pi/4$ radians, whose sine and cosine values are $\frac{\sqrt{2}}{2}$ and $\frac{\sqrt{2}}{2}$ respectively.
Thus, for $\frac{5\pi}{4}$:
$\sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$
$\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$
Answer 2
Emily Hall
To find the values of $sinleft(frac{5pi}{4}
ight)$ and $cosleft(frac{5pi}{4}
ight)$, note that the angle $frac{5pi}{4}$ is located in the third quadrant of the unit circle.
In the third quadrant, both sine and cosine values are negative.
The reference angle for $frac{5pi}{4}$ is $frac{pi}{4}$, and the sine and cosine of $frac{pi}{4}$ are $frac{sqrt{2}}{2}$.
Therefore, for $frac{5pi}{4}$:
$sinleft(frac{5pi}{4}
ight) = -frac{sqrt{2}}{2}$
$cosleft(frac{5pi}{4}
ight) = -frac{sqrt{2}}{2}$
Answer 3
Daniel Carter
The angle $frac{5pi}{4}$ radians is in the third quadrant of the unit circle.
Both sine and cosine are negative in the third quadrant.
The reference angle is $frac{pi}{4}$, with sine and cosine values of $frac{sqrt{2}}{2}$.
Therefore:
$sinleft(frac{5pi}{4}
ight) = -frac{sqrt{2}}{2}$
$cosleft(frac{5pi}{4}
ight) = -frac{sqrt{2}}{2}$
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