Calculus using series.?

Question

Use the Maclaurin series for sin x to compute sin(2°) correct to five decimal places.

Answer 1

sin(x) = x - (x^3)/6 + (x^5)/120 -+...

2° = 2*π/180 radians ≈ 0.03490658504

So sin(2°) ≈ 0.03490658504 - 0.00000708877 = 0.03489949627

It turns out you need only the first two term of the series. The Remainder Theorem tells us that the magnitude of the error from stopping after the second term is less than the magnitude of the third term. For x=0.03490658504, x^5/120 ≈ 4.3*10^(-10), so

sin(2°) = 0.03490 correct to five decimal places. (The "true" value to 11 places is 0.03489949670)

The second term of the series is less than 10^(-5) in magnitude, but you need it to correctly round to the desired accuracy.