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The Unit Circle

Overview

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.

History

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

Key Idea

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

Practice This Topic

Concept Guide

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.

Real-world example: 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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Common Pitfall

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

Word Problem
"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?"
Reasoning: 90 degrees is straight up. On the unit circle, straight up is (0, 1). So Cos(90)=0 and Sin(90)=1.

Practice Examples

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