Unit Circle

What is the unit circle?

The unit circle is the circle of radius 1 centered at the origin. Draw a ray from the center at an angle θ and it hits the circle at one special point — and the coordinates of that point are the cosine and sine of the angle: (cos θ, sin θ). That single idea ties angles, coordinates, and the trig functions together in one picture.

Degrees vs. radians

The same angle can be measured two ways. Degrees split a full turn into 360 equal parts. Radians measure the angle by the distance you travel along the circle itself: a full turn is 2π radians, so 180° = π, 90° = π/2, and 45° = π/4. Radians are the natural language of higher math because the arc length and the angle are the same number when the radius is 1. This tool always shows both so you can build the connection.

Sine and cosine are coordinates

Drop a straight line from the point down to the x-axis and you get a right triangle whose hypotenuse is the radius (length 1). The horizontal leg is cos θ and the vertical leg is sin θ — shown here in two colors. Because the hypotenuse is 1, this is exactly SOH-CAH-TOA with the "1" hidden underneath. Watch the legs flip sign as the point moves between the four quadrants, and see why tan θ = sin θ ÷ cos θ is undefined wherever cosine is 0.

How to use it

• Drag the point around the circle, or switch to Snap to lock onto the classic angles.
• Tap a common angle (30°, 45°, 60°, 90°, …) to jump there and read its exact values, like √2/2 or √3/2.
• Step or nudge the angle with the degree stepper to explore in between.
• Press play to sweep once around the whole circle and hear each common angle called out.

Good for

High-school trigonometry and precalculus (roughly grades 10–12): learning radians, memorizing the special angles, understanding the signs of sine and cosine by quadrant, and seeing where tangent blows up. A clear stepping stone toward graphing sine and cosine waves.