Use an appropriate Taylor polynomial about 0 and the Lagrange Remainder Formula to approximate sin (4/7) with an error less than 0.0001.

For a taylor polynomial

f (x) = Sum a_n x^ (n)

= Pn + epsilon

epsilon = Sum a_n+1 (x-x_0) ^ (n+1)

Lagrange remainder

a_n = f^ (n) (x_0) / n!

|epsilon| < ∫ [x_0 to x] f^ (n+1) (t) / (t - x_0) ^n / n! dt

< max (|f^ (n+1) |) (x-x_0) ^n+1 / n+1!

In the Taylor expansion of sin x

sin x = x - x^3/3! ...

find n such that epsilon < 0.0001

epsilon < max (|f^ (n+1) |) (x-x_0) ^n+1 / n+1! < 0.0001

for sin x, max (|f^ (n+1) |) < 1

(6/7) < 1 so (6/7) ^n <1

1 / (n+1) ! < 0.0001

n+1! > 0.0001

(6/7) ^7 / (n+1) ! < 0.0001

the coefficient of x^6 = 0 in the expansion of sin x

sin x = x - x^3 + x^5 + - 0.0001