The Unit Circle & Radians

Right triangles only handle angles up to 90°. The unit circle handles all of them.

Draw a circle of radius 1 centred at the origin. For any angle θ measured anticlockwise from the positive x-axis, the point on the circle is exactly (cos θ, sin θ).

That definition works for any angle, including obtuse, negative and beyond 360°.

Radians are the second way to measure angles. Instead of 360° in a circle there are 2π radians, because the circumference of a unit circle is 2π. So 180° = π, 90° = π/2, 60° = π/3.

Convert by multiplying: degrees to radians, multiply by π/180. The other way, multiply by 180/π.

Worth memorising: the values at 0, π/6, π/4, π/3, π/2.

Defining cos θ and sin θ as the coordinates of the point at angle θ on the unit circle extends the ratios to all real θ and makes both functions periodic with period 2π. The Pythagorean identity cos²θ + sin²θ = 1 is then simply the circle's equation x² + y² = 1.

Sign by quadrant follows from the coordinates: both positive in QI, sin positive in QII, both negative in QIII, cos positive in QIV.

A radian is the angle subtending an arc equal to the radius, so θ = s/r — a dimensionless ratio, which is why radians make the arc length and sector area formulas simplify to s = rθ and A = ½r²θ.

Radians are not merely an alternative convention. The derivative relationships d/dx(sin x) = cos x and the limit lim_{x→0} sin(x)/x = 1 hold only in radians, which is why all analytic work uses them.

Reference angles reduce any angle to its acute equivalent, so the memorised first-quadrant values plus a sign determine every exact value on the circle.