The Unit Circle

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.

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.

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:

These values come from the 30-60-90 and 45-45-90 special right triangles.

All Four Quadrants

Remember which functions are positive in each quadrant with "All Students Take Calculus":

• Q I: All positive
• Q II: Only sin positive
• Q III: Only tan positive
• Q IV: Only cos positive

Reference angles and quadrant signs let you evaluate trig expressions for any angle.

Beyond the Circle

The unit circle definition extends to:

• 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.