Unit Circle: Interactive Visualization

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The unit circle in this visualization

The unit circle is the circle of radius 1 centred at the origin, $x^2 + y^2 = 1$ . An angle $\theta$ is measured from the positive x-axis, counter-clockwise when positive; it ends at a point P of the circle, and cosine and sine are defined as the coordinates of that point. This defines them for every angle — obtuse, negative or more than a full turn — not only for the acute angles of a right triangle. Where $\cos\theta \ne 0$ , the tangent is their quotient. The page writes each angle in radians and in degrees, $\theta = 2\pi/3 = 120^\circ$ .

The definition, and the circle's equation

P = (\cos\theta,\ \sin\theta), \qquad \tan\theta = \frac{\sin\theta}{\cos\theta}, \qquad \cos^2\theta + \sin^2\theta = 1

In the picture the two legs of the reference triangle, from O to $(\cos\theta, 0)$ and from there up or down to P, are cos θ and sin θ as directed displacements: positive to the right and up, negative to the left and down. Line OP meets the vertical line $x = 1$ at T $= (1, \tan\theta)$ , so tan θ is the slope of OP; when θ is $\pi/2 + n\pi$ , the two lines are parallel and tan θ does not exist. T lies on the ray OP when cos θ > 0 and behind O, on its backward extension, when cos θ < 0 — that is, when sec θ < 0; tan θ can then have either sign (at 210° it is √3/3). Because P lies on the circle, $\cos^2\theta + \sin^2\theta = 1$ is the circle's own equation. Mirroring P in the axes and in the origin gives the points of $\pi - \theta$ , $\pi + \theta$ and $-\theta$ : the same triangle with other signs, which is where reduction formulas such as $\sin(\pi - \theta) = \sin\theta$ come from. Angles that differ by whole turns (coterminal angles, such as $13\pi/6$ and $\pi/6$ ) end at the same point and share all their values.

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Things to notice

The exact values of the 16 special angles (multiples of 30° and 45°) come from two triangles: half an equilateral triangle for 30° and 60°, and the isosceles right triangle for 45°. At a 30°, 45° or 60° reference angle, the triangle behind the value appears automatically: it gives the absolute value (at 30°, the chord PP′ is 1 and |sin θ| is half of it), and the quadrant gives the sign. An angle typed as a fraction of π or in degrees is held exactly; closed forms such as $\sqrt{3}/2$ appear only at the special angles, and every other value is a decimal marked ≈.
With All six, U is where line OP meets $y = 1$ : cot θ is the segment of y = 1 up to U, and sec θ and csc θ are the signed lengths OT and OU — negative when T or U lies behind O, on the backward extension of OP. tan and sec do not exist at $\pi/2 + n\pi$ , cot and csc at $n\pi$ .
The scale is fixed: it is never chosen from tan's value. When T (or U) is beyond the edge of the graph, an arrow at the edge points to it and gives its position; you can still zoom, both axes alike. Past one turn the angle's arc spirals outwards to show the winding — the widening itself means nothing — and past three turns only the last turn is drawn, with the rest counted (“+4 turns”, “−2 turns”).

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Controls

The 0°, 30°, 45°, 60° and 90° value keys set the angle exactly. Near a special angle, a dragged P catches on it; drag on and it lets go. A click on a special angle's point sends P there, the shorter way round. The angle can be typed in radians (2π/3, or a decimal such as 2.0944, which stays a decimal) or in degrees (120°), from −20π to 20π; P's arrow keys step by a degree.
Presentation mode opens a full-screen view for projectors: ← → step through the special angles, H hides every value while the picture stays.

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Unit circle

The unit circle in this visualization

Things to notice

Controls

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