The Unit Circle

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One circle to define them all — the geometric foundation of every trigonometric function.

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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.

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x² + y² = 1   ⟹   cos²θ + sin²θ = 1

This is the Pythagorean identity, the most fundamental trig identity.

Radian Measure

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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)
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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

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Memorize these values — they appear constantly in math and science:

θ = 0:   (1, 0) → sin 0 = 0, cos 0 = 1
θ = π/6 (30°): (√3/2, 1/2)
θ = π/4 (45°): (√2/2, √2/2)
θ = π/3 (60°): (1/2, √3/2)
θ = π/2 (90°): (0, 1)

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

All Four Quadrants

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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.
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.
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