What the unit circle shows
The unit circle is a circle with a radius of exactly 1, centered at the origin. It turns the trigonometric functions into something you can see: for any angle θ measured counter-clockwise from the positive x-axis, the point where the radius meets the circle has coordinates (cos θ, sin θ).
Drag the point above to any angle and watch four things update together:
- The angle, in both degrees and radians.
- cos θ — the point’s horizontal (x) position.
- sin θ — the point’s vertical (y) position.
- tan θ — the ratio sin θ ⁄ cos θ, undefined wherever cos θ = 0 (within one turn, at 90° and 270°).
Why radians matter
Degrees are convenient, but most of mathematics — and all of calculus — uses radians, where a full turn is 2π instead of 360°. One radian is the angle that sweeps out an arc equal to the radius. Turning on snap to common angles shows the exact radian values (like π⁄6, π⁄4, and π⁄3) instead of decimals.
The “special” angles worth memorizing
At 30°, 45°, and 60° (and their reflections around the circle), sine and cosine take clean exact values built from √2 and √3:
- 30° (π⁄6): sin = 1⁄2, cos = √3⁄2
- 45° (π⁄4): sin = √2⁄2, cos = √2⁄2
- 60° (π⁄3): sin = √3⁄2, cos = 1⁄2
Notice the symmetry as you drag into each quadrant: the values repeat, only the signs change. That single idea — same reference angle, predictable signs — is most of what the unit circle is teaching you.
How to use this with a class
Project it, drag to an angle, and ask students to predict the sign of sine and cosine before you cross into the next quadrant. It’s also free to embed on your own site or LMS using the snippet below.