Pythagoras on the Circle

Lesson 7 of 11 · 7 min · Free, no account needed

What you will learn

On the unit circle the legs are cos θ and sin θ and the hypotenuse is 1, so sin²θ + cos²θ = 1 for every θ.
Know one of them and the quarter, and the identity gives the other: sin θ = 0.6 on the left means cos θ = -0.8.
Games keep a direction as (cos θ, sin θ) so that its length is always 1.

Pick any angle at all. sin² + cos² is always the same number.
Which number, and why? Bet first.
Square the two legs and pour them into the tank.
Drag the point. Yellow is cos², blue is sin². Visit three different quarters.
Catch random moments.
The point turns by itself. Snap it three times and read the log.
It is Pythagoras on the circle.
The point at angle θ makes a right triangle: legs cos θ and sin θ, hypotenuse 1, the radius.
Pythagoras says leg² + leg² = hypotenuse², so sin²θ + cos²θ = 1. This is the Pythagorean identity: true for every angle.
Find cos from sin.
Turn the angle until each goal is met. The squares show why the other number has to be what it is.
sin θ = 0.8 and θ is in the first quarter. What is cos θ?
cos²θ = 1 - sin²θ = 1 - 0.64 = 0.36, and cos θ > 0 in the first quarter, so cos θ = 0.6.
This game ship flies too fast when it goes diagonally.
Its direction (dx, dy) should have length 1. Tap a fix for dy and watch one second of flight.
0.8² + dy² = 1 means dy² = 0.36, so dy = 0.6. The direction (0.8, 0.6) is a point on the unit circle.
Game: find the seat.
Each round gives one coordinate and a side. Drag the point there and let go.
sin²θ + cos²θ = 1, for every angle.
Make a badge: tap seats to stamp their two squares. Yours from earlier is first.

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