Determine the coordinates of a point on the unit circle given the angle $ heta = frac{pi}{4}$

To find the coordinates of a point on the unit circle given the angle $\theta = \frac{\pi}{4}$, we use the definitions of sine and cosine:

$ x = \cos(\theta) $

$ y = \sin(\theta) $

For $\theta = \frac{\pi}{4}$:

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

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

Thus, the coordinates of the point are:

$ \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) $

To find the coordinates at angle $ heta = frac{pi}{4}$ on the unit circle:

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

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

Coordinates are:

$ left( frac{sqrt{2}}{2}, frac{sqrt{2}}{2}
ight) $

At angle $ heta = frac{pi}{4}$:

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

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

Coordinates:

$ left( frac{sqrt{2}}{2}, frac{sqrt{2}}{2}
ight) $

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