Find the sine and cosine of the angle $heta$ on the unit circle when $heta=frac5pi4$

To find the sine and cosine of the angle $\theta =\frac{5\pi }{4}$ on the unit circle, we use the definitions of the trigonometric functions on the unit circle. The angle $\frac{5\pi }{4}$ is in the third quadrant.

For angles in the third quadrant, both sine and cosine are negative. The reference angle for $\theta =\frac{5\pi }{4}$ is $\frac{\pi }{4}$.

The sine and cosine of $\frac{\pi }{4}$ are both $\frac{\sqrt{2}}{2}$.

Thus:

$\sin \left(\frac{5\pi }{4}\right)=-\frac{\sqrt{2}}{2}$

$\cos \left(\frac{5\pi }{4}\right)=-\frac{\sqrt{2}}{2}$

To determine the sine and cosine of $heta=frac5pi4$:

Recognize that $heta=frac5pi4$ is an angle in the third quadrant, where sine and cosine are negative.

The reference angle is $fracpi4$, and:

$sinleft(frac5pi4ight)=-fracsqrt22$

$cosleft(frac5pi4ight)=-fracsqrt22$

For $heta=frac5pi4$, an angle in the third quadrant:

$sinleft(frac5pi4ight)=-fracsqrt22$

$cosleft(frac5pi4ight)=-fracsqrt22$

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