TwiddleLab / Courses / Math

Trig and Waves

Spin the unit circle, watch sine cast its shadow and turn waves into sound.

Advanced · 11 lessons · Free, no account needed

Best after: Shapes and Proof, Growth: Exponents and Logs

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Lessons

1. Right Triangles and Ratios (5 min): In a right triangle with angle A, sin A = opposite ÷ hypotenuse, cos A = adjacent ÷ hypotenuse, tan A = opposite ÷ adjacent.
2. The Unit Circle Clock (7 min): On a circle of radius 1, the point at angle θ is (cos θ, sin θ): across is cos, height is sin.
3. Radians: Measuring with the Radius (7 min): An angle in radians is the arc length divided by the radius: how many radius lengths the arc is.
4. Sine Is a Shadow (7 min): The height of a point turning on a circle, drawn over time, is the sine wave y = sin θ.
5. Waves: Amplitude, Period, Phase (7 min): In y = A sin(Bx + C), A is the amplitude: how far the crest rises above the middle.
6. Sound Is a Sine (7 min): A pure tone is a sine wave. Its frequency f (cycles per second, Hz) is the pitch; its amplitude is the loudness.
7. Pythagoras on the Circle (7 min): On the unit circle the legs are cos θ and sin θ and the hypotenuse is 1, so sin²θ + cos²θ = 1 for every θ.
8. Law of Sines (7 min): In any triangle, a ÷ sin A = b ÷ sin B = c ÷ sin C: each side over the sine of the angle across from it.
9. Law of Cosines (7 min): For any triangle, c² = a² + b² - 2ab cos C, where C is the angle across from c.
10. Rotation with Trig (7 min): Turning by θ counterclockwise is the matrix R(θ) = [cos θ, -sin θ; sin θ, cos θ].
11. Project: Make a Synth Note (8 min): A note at f is a sum of sine waves at f, 2f, 3f...: the harmonics.

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