Identities without variablesThe curious identity
Many of those curious identities stem from more general facts like the following:
Computing πAn efficient way to compute π is based on the following identity without variables, due to Machin:
A useful mnemonic for certain values of sines and cosinesFor certain simple angles, the sines and cosines take the form for 0 ≤ n ≤ 4, which makes them easy to remember.
MiscellanyWith the golden ratio φ:
An identity of EuclidEuclid showed in Book XIII, Proposition 10 of his Elements that the area of the square on the side of a regular pentagon inscribed in a circle is equal to the sum of the areas of the squares on the sides of the regular hexagon and the regular decagon inscribed in the same circle. In the language of modern trigonometry, this says:
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