Unit circle

Type an angle in degrees, like `150`, or in radians with π, like `5π/6`. You get the point on the unit circle, its cos, sin and tan, the quadrant and the reference angle. For the standard angles, the multiples of 30° and 45°, the values are exact, written with square roots, with the decimal alongside. Show the steps draws the angle, the point and the tangent.

The chart below shows every standard angle at once and prints on a single page. Tick Blank, for practice to print it with empty boxes to fill in.

The unit circle has a radius of 1 and its centre at the origin. Go round it anticlockwise from the positive x axis by an angle θ, and the point you reach is (cos θ, sin θ). The x coordinate is the cosine and the y coordinate is the sine. That single fact defines cos and sin for every angle, not just the ones that fit in a right triangle.

Only three sets of values need learning, for 30°, 45° and 60°: (√3/2, 1/2), (√2/2, √2/2) and (1/2, √3/2). Every other standard angle has one of them as its reference angle, the angle back to the nearest x axis, and only the signs change from quadrant to quadrant.

cos is positive on the right half of the circle and sin on the top half. So in quadrant I everything is positive, in II only sin, in III only tan, and in IV only cos. The order is easy to remember starting top right and going anticlockwise: all, sin, tan, cos.

tan θ is sin θ divided by cos θ, the slope of the radius. Where the radius meets the line x = 1, the height of that point is tan θ, which is how the tangent gets its name. At 90° and 270° the radius is vertical, cos is 0, and tan is undefined.

Ptolemy's trigonometry didn't use sines at all. In the 2nd century AD, in Book I of the Almagest, Ptolemy listed the length of the chord across a circle for every angle from ½° to 180° in steps of ½°. His circle had a radius of 60 parts. A chord is twice the sine of half the angle, so his table holds the same information as a sine table, written for a different circle.