$ ext{Find the coordinates of the point on the unit circle at } 45^{circ}$

To find the coordinates of a point on the unit circle at a given angle, we use the angle to find the cosine and sine values, which correspond to the x and y coordinates, respectively.

For an angle of $45^{\circ}$, we have:

$\cos(45^{\circ}) = \frac{\sqrt{2}}{2}$

$\sin(45^{\circ}) = \frac{\sqrt{2}}{2}$

Thus, the coordinates of the point are:

$(\cos(45^{\circ}), \sin(45^{\circ})) = \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$

To determine the coordinates on the unit circle at $45^{circ}$, we utilize the cosine for the x-coordinate and sine for the y-coordinate.

The trigonometric values for $45^{circ}$ are:

$ cos(45^{circ}) = frac{sqrt{2}}{2} $

$ sin(45^{circ}) = frac{sqrt{2}}{2} $

Therefore, the coordinates are:

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

The coordinates for $45^{circ}$ on the unit circle are:

$ left( cos(45^{circ}), sin(45^{circ})
ight) = left( frac{sqrt{2}}{2}, frac{sqrt{2}}{2}
ight) $

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