Find the coordinates of a point on the unit circle at an angle of $45^{circ}$ from the positive x-axis.

To find the coordinates of a point on the unit circle, we use the trigonometric functions sine and cosine.

The angle given is $45^{\circ}$.

Using the unit circle properties:

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

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

Therefore, the coordinates are $\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$.

The unit circle has a radius of 1, and at an angle of $45^{circ}$, we can determine the coordinates using the cosine and sine functions since they correspond to the x and y coordinates respectively.

Thus, we calculate:

$x = cos 45^{circ} = frac{sqrt{2}}{2}$

$y = sin 45^{circ} = frac{sqrt{2}}{2}$

The coordinates are $left( frac{sqrt{2}}{2}, frac{sqrt{2}}{2}
ight)$.

At $45^{circ}$ on the unit circle:

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

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

Coordinates: $left( frac{sqrt{2}}{2}, frac{sqrt{2}}{2}
ight)$.

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