Unit Circle Chart

The unit circle chart shows the position of the points along the unit circle that are formed by dividing the circle into eight and twelve parts. The coordinates of each point can be solved for using the corresponding special triangle.

The unit circle chart shows the positions of the points along the circle that are formed by dividing the circle into eight and twelve parts respectively.

Building the Chart

Draw a circle of radius one at the center of the cartesian coordinate plane. Label the four points where the circle intersects the x and y axis.

Unit Circle Bare

Divide Unit Circle by Eight

Divide the circle into eight equal parts and label the angle corresponding with each point. On the left, the angles are measured using radians, where one full rotation in radians is equal to 2π. On the right, the angles are measured using degrees, where one full rotation is 360°. By convention, angles are measured from the right-most edge of the circle and the positive direction is counter-clockwise.

Unit Circle Divided by Eight Unit Circle Divided by Eight

To find the position of the first point, start by observing that the 45-45-90° special triangle and the coordinates of the first point at the angle (45 °) are similar triangles.

The 45 45 90 degree triangle on the unit circle. The 45 45 90 degrees special right triangle.

Use the properties of similar triangles to solve for the x and y sides of the of the triangle formed by the point.

This gives us the (x,y) position of the point first point along the unit circle.

Point corresponding to 45 degrees on the unit circle.

Now that we have found the position of one point, we can use symmetry to fill in the other three points that correspond with the 45-45-90° triangle. This is the same as reflecting the point’s triangle horizontally over the y-axis, vertically over the x-axis, and diagonally over both axes.

The point (x,y) can be reflected across the x-axis to get the point (x,-y). The point (x,y) can be reflected across the y-axis to get the point (-x,y). The point (x,y) can be reflected across the x-axis and y-axis to get the point (-x,-y).

Filling in the these points gives us the rest of the positions formed from dividing the unit circle into eight parts.

The Unit Circle Divided by Eight

Note, some mathematicians prefer not to have a radical in the denominator and so they rewrite the fraction as:

Divide Unit Circle by Twelve

Now, we can repeat the same process for the next set of points. Start by dividing the circle into twelve equal parts and label the angles corresponding with each point.

Unit Circle Divided by Twelve Radians Unit Circle Divided by Twelve Degrees

Observe that the 30-60-90° triangle and the triangle formed by the coordinates of the first point at (30°) are similar triangles.

The 30 60 90 degree triangle on the unit circle. The 30 60 90 degrees special right triangle.

Use the properties of similar triangles to solve for the x and y component of the point.

This gives us the first point at radians.

Point corresponding to 45 degrees on the unit circle.

Then we can take advantage of the symmetry within the first quadrant of the coordinate plane to find the position of the next point associated with the 30-60-90° triangle. This point is the first triangle reflected diagonally.

Reflecting a point within the same quadrant so the point (x,y) becomes (y,x).

Then we can use the same symmetry as before to find the positions of the other points corresponding to the 30-60-90 special triangle.

Reflect Unit Circle Point over the Y axis. Reflect Unit Circle Point over the X axis.

Here is a summary of how the points are reflected during the contruction of the chart: The three points in the first quadrant are reflected accross the y-axis to get the points in the second quadrant. Then, the points in the first and second quadrant are reflected across the x-axis to get the rest of the points.

Reflect Unit Circle Points over the Y axis. Reflect Unit Circle Points over the X axis.

Unit Circle

We have finished constructing the unit circle chart demonstrating some of the properties of the unit circle.

Unit Circle

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