Unit Circle Table

The table below shows the points formed on the unit circle from dividing the circle into 8 equal parts and 12 equal parts. A point is denoted as and corresponds to the angle denoted with the variable (theta) that is formed on the circle with a radius . Each point forms the shape of a right-triangle and demonstrates the return value of the functions sine and cosine[1].

Building the Table

It makes sense to build the table in two steps. 1) Divide the circle by 8 and find the corresponding points. 2) Then divide the circle by 12 and find the corresponding points. Finally, combine the results to form the full table.

Divide the Unit Circle by 8

Start by dividing the unit circle into 8 equal parts and then start filling points corresponding to each angle. It can be helpful to write out radians in terms of fractions of a full rotation of radians and simplify if needed.

The values in the table correspond to the chart below. Note, each of the points corresponding to the angles , , and share the same right triangle geometry. The lengths of this right triangle can be solved for using the properties of similar triangles and the dimensions of the special 45 45 90 triangle.

The Unit Circle Divided by 8

Divide the Unit Circle by 12

Next, divide the unit circle into 12 equal parts and then start filling points corresponding to each angle. It can be helpful to write out radians in terms of fractions of a full rotation of radians and simplify if needed.

The values in the table correspond to the chart below. Note, each of the points corresponding to the angles , , , … share the same right triangle geometry. The lengths of this right triangle can be solved for using the properties of similar triangles and the dimensions of the special 30 60 90 triangle.

The Unit Circle Divided by 12

Combining the Tables

Combining the values of both tables results in the table at the top of the page. The values in the table correspond to the unit circle chart shown below.

Unit Circle Point

Good Links

Unit Circle | Concept

The unit circle is a unifying idea in mathematics that connects many useful concepts together. This article goes over the basic properties of the circle using interactive examples and explains how they connect to the trigonometric functions and pythagorean theorem.

Unit Circle Chart | Concept

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

Unit Circle | Interactive

This interactive demonstrates the connection between the unit circle and the trigonometric functions sine, cosine and tangent.

Radians | Concept

Radians are a unit that measure angle using the radius of a circle. One radian is equal to the amount of rotation required to travel the length of one radius along the circumference of the circle.

Degrees | Concept

Degrees is a unit of measure for angles. A full rotation is equal to 360 degrees. In the cartesian coordinate system, degrees are measured starting from the rightmost edge of the circle.

Unit Circle Chart | Resource

Printable unit circle chart. Download the unit circle chart pdf.

Blank Unit Circle Chart | Resource

Printable blank unit circle chart. Download the unit circle chart pdf.

References

  1. Unit Circle | Interactive