One circle to define them all — the geometric foundation of every trigonometric function.
The Unit Circle Definition
The unit circle is a circle of radius 1 centered at the origin. For any angle θ, the point on the unit circle is (cos θ, sin θ). This extends the right triangle definitions to all angles — not just acute ones.
x² + y² = 1 ⟹ cos²θ + sin²θ = 1
This is the Pythagorean identity, the most fundamental trig identity.
Radian Measure
A radian is the angle subtended by an arc equal in length to the radius. One full revolution = 2π radians.
Degrees to radians: θ_rad = θ_deg × (π/180) Radians to degrees: θ_deg = θ_rad × (180/π)
Radians are the natural unit for calculus: the derivative d/dx sin(x) = cos(x) only works when x is in radians. They also simplify the arc length formula: s = rθ.
Key Angles
Memorize these values — they appear constantly in math and science:
Trigonometric graphs: The sine wave y = sin(x) is the y-coordinate of a point moving around the unit circle — see applications.
Complex numbers: Euler's formula e^(iθ) = cos θ + i sin θ lives on the unit circle in the complex plane.
Polar coordinates: Every point in the plane as (r, θ) — extending the circle to all radii. See the main trigonometry page.
The unit circle connects geometry, algebra, and calculus in a single picture. It's the Rosetta Stone of mathematics — learn it well, and all three subjects become clearer.