The diagram attached shows the graph described. Remember that the radius of the Unit Circle is 1. The In the corner of the diagram, we have the special right triangle 30-60-90 degrees, and its unit length ratios.
Using the ratios, we know that the side opposite the 30° angle (the x-axis) will be 1/2 the hypotenuse 1, which is 1/2. This means our x-coordinate is 1/2.
We also see that the side opposite the 60° angle will be the other leg, times √2. This means it will be (1/2) * √2 = √(2)/2. This is the y-coordinate.
Therefore, the coordinates of the point on the unit circle are (1/2, √(2)/2).