In This Lesson The Unit Circle Definition Radian Measure Key Angles All Four Quadrants 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 Beyond the Circle 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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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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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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.