Unit Circle

Why the unit circle is the heart of trigonometry

The unit circle is a circle of radius 1 centered at the origin, and it is the single most useful picture in trigonometry. Pick any angle θ measured counter-clockwise from the positive x-axis, and the point where the angle's ray crosses the circle has coordinates exactly (cos θ, sin θ). That is the whole idea: the horizontal coordinate is the cosine, the vertical coordinate is the sine. This tool draws that picture live — drag the blue point around the circle, or use the slider, and watch the coordinates, the angle, and the sine, cosine, and tangent values update together. Everything is drawn on a canvas in your browser.

Reading the diagram

When you turn on the reference triangle, the tool draws the right triangle formed by the radius, the x-axis, and a vertical drop from the point. Its horizontal leg (red, dashed) has length cos θ; its vertical leg (green, dashed) has length sin θ. Because the radius (the hypotenuse) is 1, the classic ratios from right-triangle trigonometry — sine = opposite/hypotenuse, cosine = adjacent/hypotenuse — simplify to just the leg lengths. The small arc near the center shows the angle being swept, and the label by the point reads off the exact (cos, sin) coordinates to three decimals.

A worked example at 45°

Set the angle to 45°. The point sits at the top-right of the circle, and the diagram reads (0.707, 0.707). That is no accident: at 45° the reference triangle is isosceles, so its two legs are equal, and each equals √2 / 2 ≈ 0.7071. The tool shows the radian equivalent as π/4 rather than a decimal, because 45° is a special angle. Tangent, which is sin θ / cos θ, equals 0.707 / 0.707 = 1 here. Drag on to 90° and watch cosine fall to 0 while sine rises to 1; tangent is then reported as undefined, because dividing by a cosine of zero has no finite value.

Special angles and exact values

The tool knows the 16 standard angles around the circle — 0°, 30°, 45°, 60°, 90° and their counterparts in every quadrant — and marks them as small dots. Snap to any of them and the radian label switches from a decimal to its exact symbolic form. These are the values worth memorizing:

Notice the symmetry: the sine and cosine of 30° and 60° are mirror images, and the whole table repeats with sign changes in the other three quadrants. Watching the signs flip as you drag the point between quadrants is the fastest way to internalize that pattern.

How the angle is measured from your drag

When you click or drag, the tool converts your pointer position into an angle with atan2(y, x), the standard function for turning a coordinate into a direction. Because screen y grows downward but math y grows upward, it flips the vertical sign first, so dragging up gives a positive angle as you would expect. The angle is then normalized into the 0–360° range for display.

Common use cases and study tips

• Learning the sign rules. Drag through all four quadrants and note when sine and cosine are positive or negative — the "all, sin, tan, cos" pattern becomes obvious visually.
• Connecting degrees and radians. Use the special-angle snap buttons to pair each degree value with its exact radian form until the conversion is automatic.
• Seeing why tangent blows up. Approach 90° or 270° and watch the vertical leg stretch while cosine shrinks toward zero — a concrete picture of the tangent asymptote.
• Teaching. The reference triangle and projections make a clear demonstration of why sin² θ + cos² θ = 1: it is just the Pythagorean theorem on a hypotenuse of length 1.
• Common mistake: forgetting that the slider and most calculators work in different angle modes. Here the slider is in degrees while the radian value is shown alongside, so you always see both — don't mix them up when plugging values into a calculator set to radians.