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 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 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 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.