The Unit Circle

A Special Circle

The unit circle is a circle with:

• Radius = 1
• Center at the origin (0, 0)

That’s it. Nothing more.

Why Radius 1?

Because it makes everything simpler.

When the radius is 1, the coordinates of any point on the circle give us the values of cosine and sine directly.

No division. No extra steps. Just read off the coordinates.

The Key Insight

Take any point P on the unit circle. Call the angle θ.

The coordinates of P are:

P = (cos θ, sin θ)

The x-coordinate is the cosine. The y-coordinate is the sine.

Why Does This Work?

For now, take this as a definition: cosine and sine are the x and y coordinates of a point on the unit circle.

Later, when we study right triangles, we’ll see why this definition makes sense and connects to triangle ratios.

Beyond Right Triangles

With right triangles alone, angles are limited to between 0° and 90°.

The unit circle removes that limitation.

Any angle works:

• Negative angles? Just go clockwise.
• Angles over 360°? Keep wrapping around.
• Angles like 150° or 240°? The circle handles them all.

The terminal side hits the circle somewhere, and that point’s coordinates give you cosine and sine.

Key Points

Some angles give especially clean values:

These four points are easy to remember: they’re where the circle crosses the axes.

The Pythagorean Identity

Every point on the unit circle satisfies:

$x^2 + y^2 = 1$

Since $x = \cos \theta$ and $y = \sin \theta$:

$\cos^2 \theta + \sin^2 \theta = 1$

This is the Pythagorean identity, and it’s one of the most important equations in trigonometry.

It’s just the equation of the unit circle, written in terms of sine and cosine.