To put some numbers on it, using N terms of the Taylor series for sin(x) [1] with |x| <= 0.1, the maximum error percentage cannot exceed [2]:
N Error Limit
1 0.167% (1/6%)
2 8.35 x 10^-5% (1/11980%)
3 1.99 x 10^-8% (1/50316042%)
4 2.76 x 10^-12% (1/362275502328%)
Even for |x| as large as 1 the sin(x) = x approximation is within 20%, which is not too bad when you are just doing a rough ballpark calculation, and a two term approximation gets you under 1%. Three terms is under 0.024% (1/43%).
For |x| up to Pi/2 (90°) the one term approximation falls apart. The guarantee from the Taylor series is within 65% (in reality it does better, but is still off by 36%). Two terms is guaranteed to be within 8%, three within 0.5%, and four within 0.02%.
Here's a quick and dirty little Python program to compute a table of error bounds for a given x [3].
[1] x - x^3/3! + x^5/5! - x^7/7! + ...
[2] In reality the error will usually be quite a bit below this upper limit. The Taylor series for a given x is a convergent alternating series. That is, it is of the form A0 - A1 + A2 - A3 + ... where all the A's have the same sign, |Ak| is a decreasing sequence past some particular k, and |Ak| has a limit of 0 as k goes to infinity. Such a series converges to some value, and if you approximate that by just taking the first N terms the error cannot exceed the first omitted term as long as N is large enough to take you to the point where the sequence from there on is decreasing. This is the upper bound I'm using above.
For |x| up to Pi/2 (90°) the one term approximation falls apart. The guarantee from the Taylor series is within 65% (in reality it does better, but is still off by 36%). Two terms is guaranteed to be within 8%, three within 0.5%, and four within 0.02%.
Here's a quick and dirty little Python program to compute a table of error bounds for a given x [3].
[1] x - x^3/3! + x^5/5! - x^7/7! + ...
[2] In reality the error will usually be quite a bit below this upper limit. The Taylor series for a given x is a convergent alternating series. That is, it is of the form A0 - A1 + A2 - A3 + ... where all the A's have the same sign, |Ak| is a decreasing sequence past some particular k, and |Ak| has a limit of 0 as k goes to infinity. Such a series converges to some value, and if you approximate that by just taking the first N terms the error cannot exceed the first omitted term as long as N is large enough to take you to the point where the sequence from there on is decreasing. This is the upper bound I'm using above.
[3] https://pastebin.com/thN8B7Gf