In This Lesson The Unit Circle Definition Radian Measure 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 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.
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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.
Key Angles 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 Remember which functions are positive in each quadrant with "All Students Take Calculus":
Q I: All 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 Q II: Only sin positive Q III: Only tan positive4/ddvb2ENoXKjP3or4fVlbahYOP3FSbGWZNXP6qv0m250ATdtz6RO++yhJSxkWzekP/lEcDqBOKpRufmnc+Ej39+wZXRorWBvuAEajMAAcehn6qw/MTH0YSPlhk2qx9qlwZpFTU4Ocjkxn9o4vkB0P02sP6ji/iJkguDXd4k/74McVbYj2G046euZyKZd3W6Fta0tbfSewlN9KcalNGMv32fNsLBUHMKMRetNXJnP/Ro50RvIJEKDjgXA5VzknJd9wLp58mG/ixnPj4PnmiUFPzLyx0wJkAixB0mfKmn0Z2oi7TGEkTG1Ak/eduoEuPPBZcqvnDlhwU6B+YQNJmLf6ArHeLyYIp3XcWU0vV9HY+eVew2chMu3CX6zGYG3LqWrEn34B+RYlKPmF02vwhEld21COsn8BeRrWZhDzSgf5AKcfBVKwK9B8yttgLNjC58lZYcwblkBETse03lAgyXl0vQdk+CCAl1BEj8S2bV12vh83mwjeVXMC17mLIoKTtEx0c5VGU5U5J8h+lmYqJD0A2NRAhi7vbgjRLMzL8gXRxQifsvGQvYz6TCw6Y9n4YxCy2om4gH4LUUTuQdzDePzSeF/FMYISKC/jZsrBDpOox0E+EczOeU49EjZkUvYJvZ8VD2H8EJvTbxjDe/eDo+JAoioq4eInVkoO9hlOe5cjT30ImYBMch3zwwkSpb/6mT3Etk364ld3n 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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