OneTaskBy Tim King · Updated
Unit Circle
Exact Values in Radians and Reference Angles for Sin, Cos and Tan
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How the Interactive Unit Circle works
This interactive unit circle shows sine, cosine and tangent as real lengths. Drag the point round the circle, or type an angle, and the blue cos line, orange sin line and green tan line move with it. Every special angle is marked in degrees and radians, and the point's coordinates are given as exact values wherever they exist.
The unit circle is a circle of radius 1 centred on the origin. Any angle, measured anticlockwise from the positive x-axis, lands on one point of the circle, and that point's coordinates are (cos θ, sin θ). That single fact defines the trig functions for every angle, not just the acute ones in a right triangle.
How to read the unit circle
The blue line from the centre along the x-axis is cos θ. The orange line straight up or down to the point is sin θ. Together with the radius they form a right triangle with hypotenuse 1, which is why sin² θ + cos² θ = 1. The green line touches the circle at (1, 0) and runs up to where the radius, extended, meets it; its length is tan θ.
At 30° the point is (√3/2, 1/2), so cos 30° ≈ 0.866 and sin 30° = 0.5. Drag on to 150° and the point is (−√3/2, 1/2): the same height, mirrored left, because 150° has the same reference angle.
Tips
- Snap for exact values. The 15° snap lands only on angles with exact forms. Turn it off to explore in-between angles.
- Watch the signs. The point's x and y signs tell you which functions are positive in each quadrant.
- Tan runs off. Near 90° and 270° the tangent line grows without limit, so it is hidden there.
FAQ
How do you memorise the unit circle?
Learn one quadrant. The first-quadrant points are (1, 0), (√3/2, 1/2), (√2/2, √2/2), (1/2, √3/2) and (0, 1). Every other special point is one of these with the signs changed for its quadrant. A neat pattern: the sines at 0°, 30°, 45°, 60° and 90° are √0/2, √1/2, √2/2, √3/2 and √4/2.
What are the radian values on the unit circle?
A full turn is 2π, so 90° is π/2, 180° is π and 270° is 3π/2. The 30° family is multiples of π/6, the 45° family multiples of π/4 and the 60° family multiples of π/3. The outer ring of labels on the circle shows each one.
Why is the unit circle radius 1?
With a radius of 1 the coordinates of the point are the sine and cosine themselves, with no scaling. On a circle of radius r the point would be (r cos θ, r sin θ), which is how polar coordinates convert to x and y.
What happens to angles above 360° or below 0°?
They wrap round. 390° lands on the same point as 30°, and −45° on the same point as 315°. Type either into the angle box and the circle shows the matching position between 0° and 360°.
See also: all Trigonometry and Calculus tools, Arithmetic, Number Theory, Algebra and Geometry, Statistics.
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