The Unit Circle

Welcome to trigonometry. Forget about complicated triangles for a moment; we are looking at a circle with a radius of exactly 1. This is the basis for everything related to sine and cosine.

1. The Basics: Radius 1

The unit circle has its center at (0, 0). The distance to the edge is 1 everywhere.

📐 The angle (α): We measure from the right x-axis, counterclockwise.
🧭 The Quadrants: The circle is divided into four planes (I to IV).

2. The Coordinates

If you place a point on the circle, the coordinates immediately tell you the sine and cosine:

🔵 The x-axis is the Cosine: x = cos(α)

🔴 The y-axis is the Sine: y = sin(α)

Why? Because the hypotenuse is 1, the formula becomes simple:

cos(α) = x1 = x
sin(α) = y1 = y

Interactive Experimentation 🕹️

In the graph below you can see how the sine graph follows from the unit circle.

3. The Four "Cardinal Directions"

You can read these values directly without a calculator. Just look at the extreme points of the axes:

Angle (α)	Coordinate (x, y)	cos(α)	sin(α)
0°	(1, 0)	1	0
90°	(0, 1)	0	1
180°	(-1, 0)	-1	0
270°	(0, -1)	0	-1