In This Lesson The Unit Circle Definition Radian Measure Key Angles All Four Quadrants Beyond the 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.
x² + y² = 1 ⟹ cos²θ + sin²θ = 1
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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.
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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)
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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 .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 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. 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
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.