Find the sine and cosine of the angle $ heta = frac{pi}{4}$ on the unit circle.

On the unit circle, the coordinates of a point corresponding to an angle $\theta$ are $(\cos\theta, \sin\theta)$. For $\theta = \frac{\pi}{4}$, we need to find the sine and cosine values.

The angle $\frac{\pi}{4}$ is 45 degrees.

Using the unit circle properties, we know:

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

and

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

So, the sine and cosine of $\theta = \frac{\pi}{4}$ are both $\frac{\sqrt{2}}{2}$.

To find the sine and cosine of $ heta = frac{pi}{4}$ on the unit circle, we use the fact that on the unit circle, the coordinates $(x, y)$ correspond to $(cos heta, sin heta)$.

For $ heta = frac{pi}{4}$:

$ x = cosleft(frac{pi}{4}
ight) = frac{sqrt{2}}{2} $

and

$ y = sinleft(frac{pi}{4}
ight) = frac{sqrt{2}}{2} $

Therefore, the cosine and sine of $ heta = frac{pi}{4}$ are $frac{sqrt{2}}{2}$ each.

The unit circle gives us $(cos heta, sin heta)$ for any angle $ heta$.

For $ heta = frac{pi}{4}$:

$ cosleft(frac{pi}{4}
ight) = frac{sqrt{2}}{2} $

and

$ sinleft(frac{pi}{4}
ight) = frac{sqrt{2}}{2} $

Thus, both the sine and cosine of $ heta = frac{pi}{4}$ are $frac{sqrt{2}}{2}$.

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