Write the Taylor series for f(x)=sin(x)

Question

Write the Taylor series for f(x)=sin(x)  at x= pi/2

as

∞

∑ cn(x- (pi/2))^n

n=0

find the first five coefficients.

Answer 1

The coefficients of Taylor series for f about x = a is

cn = fn(a) / n!, where fn is the nth derivatives.

Therefore,

f0(a) = f(a) = sin(pi/2) = 1, c0 = 1

f1(a) = f'(a) = cos(pi/2) = 0, c1 = 0

f2(a) = f''(a) = -sin(pi/2) = -1, c2 = -1/2

f3(a) = f'''(a) = -cos(pi/2) = 0, c3 = 0

f4(a) = f''''(a) = sin(pi/2) = 1, c4 = 1/4! = 1/24