Take a triangle. If you reduce it and enlarge it, the overall shape doesn’t change, right? Even as one side shrinks or grows, the other sides also shrink and grow to match.
The ratios between each of the three sides is constant and depends only on the angles between the sides. Cos/sin/tan are basically those ratios for a given angle. (Technically, they only work on triangles with a right angle, but you can make any triangle out of two of thoose, so it all works out in practice)
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