alt Hacker NewsTo construct the two vectors spanning a given angle (well, as the article says, one of them can be chosen arbitrarily, the easiest is to just use (1, 0)), you'll of course need trigonometric functions. This is exactly equivalent to how constructing a rotation matrix from an angle requires trigonometric functions, but after that you can rotate how many vectors you like with just dot products, or how constructing e^ai = sin(a) + cos(a)*i requires trigonometric functions, but after that it's just complex multiplication.
Another fun way to decompose 2D rotation is into three shears, originally given by Paeth in 1986: https://silmon.github.io/arbitrary-image-rotation-using-shea...
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Because an "angle" is by definition trigonometric, it's a ratio of a circle perimeter. You can also use an angle already formed by two vectors and won't need trigonometry. You can also use others things than trigonometric angles, such as spread from rational trigonometry
Older versions of mspaint had shear but not rotation (to arbitrary angles), and you could use the same approach to achieve rotation via shearing.