Use an appropriate Taylor series to approximate the given value, accurate to within 10^-11 .

Use an appropriate Taylor series to approximate the given value, accurate to within 10^-11 .

cos3.04

Note that 3.04 is near π.

So, we expand cos x in powers of x - π for a faster approximation.

On one hand,
cos(x - π)
= cos x cos π + sin x sin π
= -cos x

On the other hand,
cos(x - π)
= cos x cos π + sin x sin π
= -cos x
= Σ(n = 0 to ∞) (-1)^n (x - π)^(2n)/(2n)!.

Hence, -cos x = Σ(n = 0 to ∞) (-1)^n (x - π)^(2n)/(2n)!
==> cos x = Σ(n = 0 to ∞) (-1)^(n+1) (x - π)^(2n)/(2n)!.

So, cos 3.04 = Σ(n = 0 to ∞) (-1)^(n+1) (3.04 - π)^(2n)/(2n)!
------------------
Since this series is an alternating series, the error after the nth term is bounded above
by the absolute value of the (n+1)-th term.

Hence, we want (π - 3.04)^(2n+2)/(2n+2)! < 10^(-11).
==> n = 3 by trial and error.

So, cos 3.04 ≈ Σ(n = 0 to 3) (-1)^(n+1) (3.04 - π)^(2n)/(2n)!
....................≈ -0.994843903359 (to 12 decimal places)

(Double checked on Wolfram Alpha.)

I hope this helps!