In This Lesson The Unit Circle Definition Radian Measure Key Angles JVYLxteDzC60Nv2p0/5uBtmGVSAtBOPlT+ctwM5budm6yfvYd4hr6DiQTeHKyoWWpC6B+of9Ts6WwGX2CV/NPg1ItiKY45euhTvbwtbUV6z/taR1IK/szOa8uQ0K+iUwQtb707Q+T0t8zyr0PCT/ij0ydIS6JeYYLFBfefEvPbzobIG0ui9HWGndqeS6EZKlCxFDsdBunsLXLTK6Pj6FMVGeozpvS14Dc8MaNEGyRCTCsJryMpEkfGfjztc7BTZ02lJ9KKE6KPFrZz2dx54M526YqCKufoY9o4JZCQwPdg9Xx/rGdVRW1Dhj9QTOSTxfffFlOPNVZnOdUWeFypRDjAwsxzUl8p9w9s0pfCjx0UcP0dq/67lcjUrYv6rAPuFrAPqPRw6aikPzXwBFucd+zTJY/oPGX40vicHr2kaPgf00vIJjMgsuHLdF3Fei/4/NPWrbFY4cTpuQR+LPbK0+kbHQWKlOt8VhozVZC9NdQJASmNi9igDN0juGM5630E4D/UBJTIhdMuVFZzlgJ0Og10gxKAVMwH6L61/fY3uxFSmyuyQotcoresmlpQYhWdyoqc7hgsytplvFXcxTKcQqVT0JQz8SGYDbJ1T+rSgYd2IsKr46L/aVOjlh/NlRjfUWDh7FF/BF36dg/2XjXFsrkXElpbGu1HBijlEtvTwEIxAUff9GTZmr9vGtrWWWPDg7/AZM1xEf All Four Quadrants 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.
x² + y² = 1 ⟹ cos²θ + sin²θ = 1
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)
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 positive Q II: Only sin positive Q III: Only tan positive3hKzrRs6Dw/hXW19Ehl4+I//PTeMVBmms78xvnU3UfliibZkJkC2hpWQyr/xLvDgN2Ua8OdxrIg+79ziXzXfNmosJUehjOp/0QsWCHcSBTF7Q+UMzPzfGESb1SjM3MtWspMmhJurSRDRaR6DqCKYwy2+BEu+hVOIh+SejyEVFf0+mlpkTfNpG78Rbmju9hscea+WrZiSDqz4UDrQfiEzfzzfoPaCNTxpEyqJR84JhtN4r5bahTJcTYYuYq0qepqcP1BBnQ1STFnOaBB3UWbkVo/LXFXZQYCJmHeLqw2Db7wwhzQBRq8vlLGACPy8tdqwhZzv/gULyEwbfNfKP/8gvkRRTelsuZ86pSyKXh/0lW6FP2rofw0mAY++smpWRxnHFVvLbbWePkXuZbvCygTC2bzb3y9cJKSI6pi1QBEKcZsukVjwqIL1y/ttWghQWTOAAfyFpapPDPduRPbfvTqUZL+j28HudOC4kV+GwP+LM+7bke50JXRYfx1fvdKrSmI1t7/H8WKtkEardw2xmXw89chA/UttQTJIoq+VE0gFKdYUXIswpIQtmLFAiW9yq/rd2diYMSM9J8S4/QwxPGLzFThB08nnntlkDJWatJ0pqafSQF6+FBWbI+k6mEBeCqQF32H1TWbptATZeIoPOJfJ4Age9etBSFay8cJJgA3mk7UTtjr0IqZZtXAiwe8+AmbEVoz2zx0v Q IV: Only cos positive Reference angles and quadrant signs let you evaluate trig expressions for any angle.
Beyond the Circle 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 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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