Determine the coordinates of a point on the unit circle at an angle of ( frac{pi}{4} )

To find the coordinates of a point on the unit circle at an angle of \( \frac{\pi}{4} \), we use the unit circle definition:

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The unit circle is defined as all points (x, y) such that:

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$ x^2 + y^2 = 1 $

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For an angle \( \theta \), the coordinates are given by:

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$ (\cos(\theta), \sin(\theta)) $

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At \( \theta = \frac{\pi}{4} \):

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

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

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So, the coordinates are:

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

To find the coordinates of a point on the unit circle at an angle of ( frac{pi}{4} ), use the unit circle:

For an angle ( heta ), the coordinates are:

$ (cos( heta), sin( heta)) $

Thus, at ( heta = frac{pi}{4} ):

$ 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) $

For an angle of ( frac{pi}{4} ) on the unit circle, the coordinates are given by:

$ (cos( heta), sin( heta)) $

Thus, at ( heta = frac{pi}{4} ):

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

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