In This Lesson The Unit Circle Definition Radian Measure Key Angles All Four Quadrants Beyond the Circle The Unit Circle Definition 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 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 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:
θ = 0: (1, 0) → sin 0 = 0, cos 0 = 1
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 positiveQ II: Only sin positiveQ III: Only tan positiveQ IV: Only cos positive Reference angles and quadrant signs let you evaluate trig expressions for any angle.
Beyond the Circle elOHD8pzTuwrlJTs7VNZbmJFZs/m+ZIwMP2eS5nHYuTBukUxY4VNiuBmvD628IdwDKmIXnK658AaHtgKzQ06+/IJ5w7mjJc8wZElxswrifZkvCSSv5KvzHHZTW9mO5Z9PNM6UXDftMAnFHvT9mI1Q6l6/BOlNYLwiSV26KODkWHrtrrE59sm16xq3CLhOCxuJ+yveWn3gtElZiXtt6WHD+WfkTDqfASaohJeVkVp8qGaO9hud+pbITqE75bCQcW8lMSnuQfQhY31t88i9Tcxu1bPHMZlhPaSIKTe7gT4HDuGtiUrzfnsVWzXYUa1wT8euGUWKfP9WN/FviYyK6m34oC0UCorLHezD4fkOxLJOohKdrCMg9ki6vSXoLyRHmZ0by9Qk6SWNCgzkaaqdZPLuJqWVfdVDsrcz4iAV5UZ47cRUnqoxu/2/qEUOBBD9CYBLvdeRGBB2scWWUp9v7ls09NhYtwzammLQsfMSfEUoZSZqLGBaESK6C7UU6cvAioy0k0FigdGlI5dl1BswZuvE52duUDEdcpc/kvHSdgAL2w2fFz/ic6OEa5HZwRvRpAYY9JFhon35ZizYAdGm8O2ZVcEHbG4QyLoKrshjbDyBw3cErnWeujn8K/5lDoDq/JA4YL9KkOnN7UDpLlkLVMjPmWIiQtGItrcggSS4oBu9HRcGrBlLIJnYaQxKT2gCnosmMHrbDSOF 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. 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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.