The Unit Circle Clock

Trigonometry · Axiom Academy

See how a 24-hour clock transforms into the unit circle—where every hour maps to an angle, and every position reveals trigonometric secrets! Have you ever noticed that a clock face is just a circle? And circles are at the heart of trigonometry! Let's discover how telling time can teach us about angles and the unit circle. Just like a clock hand sweeps around the clock face, angles sweep around the unit circle. Let's explore this connection! Use the slider to set different times on a 12-hour clock. Watch how each hour position corresponds to an angle on the unit circle! Every position on the clock (and unit circle) has coordinates (x, y). These coordinates are exactly cos(θ) and sin(θ) ! At 3:00 , we're at 90° (π/2 radians). Here, cos(90°) = 0 and sin(90°) = 1. This is why the point is at the very top of the unit circle—all the way "north" with no "east-west" component! You've explored different times and seen how the angles and coordinates change. Let's test your understanding! cos(30°) ≈ 0.866 sin(30°) = 0.500 cos(330°) ≈ 0.866 sin(330°) = -0.500 What do you notice about cos(30°) and cos(330°)? Unit Circle Clock: Key Insights Clock = Circle = Angles Each hour position is 30° (π/6 radians) from the last Coordinates = Trig Functions x = cos(θ) and y = sin(θ) at any point on the unit circle Symmetry Patterns Angles in different quadrants share trig values (with sign changes) Pythagorean Identity sin²(θ) + cos²(θ) = 1 always holds on the unit circle

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