Constructing a unit circle


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In trigonometry, mastering the unit circle is a precondition for success. Loaded up with angles, sines, cosines, and tangents, it looks at first like a mess. But it can be built up in a careful orderly manner. The circle is loaded with patterns, so if you learn how to deal with , , and , you know almost everything.

There is a tidy way of constructing the unit circle from a square. The order of the steps can be varied, but here is a unit circle handout where the user can follow these of steps:

  1. draw a circle of radius one
  2. circumscribe a square with sides of length 2
  3. subdivide the square into 4 smaller squares (sharing a vertex at the center of the circle)
  4. subdivide each of these squares into 4 squares (for a total of 16)
  5. lines from the origin to the vertices of the various squares create all of the standard angles in a unit circle: , , , , , , , ..., , .

Depending on the application, you likely prefer radians.

A full-page unit circle helps reinforce the patterns in the unit circle. I recommend that students put in coordinates for all points, fill in all 16 angles, and consider showing the tangent value for each angle where the tangent is defined.

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