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The Unit Circle

The Hook

The Circle of Life.

What is it?

A circle with a radius of 1, centered at (0,0), used to define trig functions for all real numbers.

Time Travel (History)

Popularized by Euler to connect algebra, geometry, and complex numbers.

Cheat Code

Cos is X, Sin is Y. (Alphabetical order: C before S, X before Y).

Practice This Now!

The Field Guide

In Plain English: Imagine a clock hand that is exactly 1 foot long. As it spins, the tip traces a circle. At any moment, the distance the tip is from the center-line (left/right) is the Cosine. The height of the tip (up/down) is the Sine.

In The Real World: Robotics. Robotic arms use the unit circle logic to determine exactly where the 'hand' is based on the rotation of the 'elbow'.

How To Do It

  1. Draw a circle with radius 1 on a graph.
  2. Draw a line from the center to the edge at your angle.
  3. Drop a line down to the X-axis to make a right triangle.
  4. The base is Cos(angle), the height is Sin(angle).
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Booby Trap!

Forgetting the signs! In Quadrant II (top left), X (Cos) is negative but Y (Sin) is positive.

Real World Challenge
"A Ferris wheel has a radius of 1 unit. You get on at the far right (position 1,0). You rotate 90 degrees counter-clockwise. What are your coordinates?"
The Logic: 90 degrees is straight up. On the unit circle, straight up is (0, 1). So Cos(90)=0 and Sin(90)=1.

Training Drills

Coordinates
(0,1) | (-1,0)+---(1,0) | (0,-1)
The four cardinal points of the Unit Circle.