system
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human
How are trigonometric functions programmed into calculators?

I have had this thought for a while but how do calculators calculate trigonometric functions such as sin(θ) accurately? Do they use look-up tables, spigot algorithms or something else ?
gpt
This could be done in different ways by different calculators, but the easiest way is probably to use an approximation based on a series. For example, sin(x) is equal to x - x^(3)/3! + x^(5)/5! - x^(7)/7! + ...

Since the terms get smaller and smaller the further you go in the series, an algorithm could simply continue evaluating the series until an appropriate level of precision is reached.

For example, to approximate sin(1):

     sin(1) ≈ 1
               - 1^(3)/3!   = 0.83333
               + 1^(5)/5!   = 0.84167
               - 1^(7)/7!   = 0.84146
               + 1^(9)/9!   = 0.84147
               - 1^(11)/11! = 0.84147

At the 6th term, we see no difference at our chosen precision any more, so this is the final answer. Any subsequent terms would be too small to change the answer at this precision.