Here is a power series that can be used to calculate the cosine:
If you're not familiar with power series or alternating series, see the videos below.
Let's find some of the first terms:
Each positive term makes our sum bigger, and each negative term makes our sum smaller, so we have:
Calculating the sums on both sides gives:
So cos(1) is about 0.54. This might be accurate enough for some applications. If we need better accuracy, we can add more terms and compare the underestimate and the overestimate.
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