Sine and cosine are just the shadows of a point moving steadily around a circle of radius 1: the shadow on the vertical axis is sine, on the horizontal axis cosine, and because the radius is 1 the Pythagorean theorem immediately gives sin²θ + cos²θ = 1. Cosine always runs a quarter turn ahead of sine, while tangent is the length of the tangent segment erected at (1, 0), which is why it races off to infinity every time θ reaches 90 degrees and cosine hits zero. Let the two coordinates oscillate at frequencies in a 3:2 ratio and the pen traces a knotted Lissajous figure.
Sine and cosine are just the shadows of a point moving steadily around a circle of radius 1: the shadow on the vertical axis is sine, on the horizontal axis cosine, and because the radius is 1 the Pythagorean theorem immediately gives sin²θ + cos²θ = 1. Cosine always runs a quarter turn ahead of sine, while tangent is the length of the tangent segment erected at (1, 0), which is why it races off to infinity every time θ reaches 90 degrees and cosine hits zero. Let the two coordinates oscillate at frequencies in a 3:2 ratio and the pen traces a knotted Lissajous figure.
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