Walk the Unit Circle

What to notice

The radius is 1, so the hypotenuse of the triangle is 1. Then cos θ = adjacent/1 and sin θ = opposite/1: the legs are the cosine and the sine, and the point is (cos θ, sin θ). By the Pythagorean theorem, cos²θ + sin²θ = 1 for every angle.

One full turn is 360°, or 2π radians, because the circle's circumference is 2π·1. An angle of θ radians cuts off an arc of length θ on the unit circle.

Reference angles and signs

The reference angle θ′ is the acute angle between the terminal side and the x-axis. Angles with the same θ′ have the same cosine and sine except for sign, and the sign comes from the quadrant: All, Sine, Tangent, Cosine are positive in Quadrants I, II, III and IV.

On an axis (0°, 90°, 180°, 270°) there is no reference angle, and tan θ is undefined at 90° and 270° because x = 0 there.

Hipparchus and Ptolemy built tables of chords in a circle almost two thousand years ago; the sine came later through Indian and Arabic astronomers, and in 1748 Euler wrote them as functions on a circle of radius 1, as we do here.

Try it

1. Find all angles between 0° and 360° with sin θ = 1/2. What is their reference angle?
2. Go to 135°, 225° and 315°. Which coordinates stay the same size, and which signs change?
3. Turn slowly from 80° to 100°. What happens to tan θ, and why?
4. Where on the sine graph is the point when θ is in Quadrant III?
5. Find θ (other than 45°) where cos θ = sin θ.