You can use a Taylor series to approximate common functions. The Taylor series for sin(x) is
Using only the first several terms in the series could get you a good approximation to the function.
Write a function that takes a point x at which to evaluate the sine function and a tolerance level that defines how close you want the Taylor series approximation to be to the actual value. Your function should output the approximate value of sin(x).
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Using sin(x) to test the tolerance level is probably inadequate. We should be testing how much a new term improved the solution compared to the previous one. If we calculate sin(x) there is no need to use the Taylor series.
You are right. I made a solution with the remainder of Lagrange for error estimation. I avoided the sin(x) usage for comparing approximation and real values.