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 one of the two corresponding special triangles.

The unit circle chart shows the positions of the points along the unit circle that are formed by dividing the circle into eight and twelve parts respectively. This figure is measured using radians and degrees.
Figure 1: Unit Circle Chart π (pi)

Building the Chart

Draw a circle of radius one at the center of the cartesian coordinate plane. Make sure to draw it big enough to leave space for all of the points or print out a blank unit circle chart. Label the four points where the circle intersects the and axis.

This figure illustrates the points where the unit circle chart intersects the x and y axis in the cartesian coordinate system.
Figure 2: Unit Circle Chart Intersections

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 is equal to . On the right, the angles are measured using degrees, where one full rotation is . 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 () 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 and sides of the of the triangle formed by the point.

This gives us the position of the first point along the unit circle. Note, mathematicians prefer not to have a radical in the denominator and so they rewrite the fraction: .

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 horizontally over the -axis, vertically over the -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 chart shows the positions of the points along the unit circle that are formed by dividing the circle into eight and twelve parts respectively.
Figure 3: Unit Circle Chart Eights π (pi)

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 coordinates of 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 is visualized by reflecting the triangle diagonally.

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

Like before, we can reflect the position of a point over the axes to find the other points corresponding to that point. Follow this process for each of the two points corresponding to the 30-60-90° triangle in the first quadrant:

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 these points gives us all the positions of the points formed by dividing the unit circle by as shown in figure 4.

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.
Figure 4: Unit Circle Chart Twelfths π (pi)

Then, when we combine the points found from dividing the circle by with these points, we have finished constructing the unit circle chart.

Unit Circle Chart

In conclusion, the unit circle chart demonostrates some properties of the unit circle. It results from dividing the circle into and sections respectively. Each point from the divisions corresponds to one of the two special triangles: 45 45 90 triangle and 30 60 90 triangle.

The unit circle chart shows the positions of the points along the unit circle that are formed by dividing the circle into eight and twelve parts respectively. This figure is measured using radians and degrees.
Figure 5: Unit Circle Chart π (pi)

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