We need to find the first, second, third, etc derivatives and evaluate them at x = 0. Starting with:

f(x) = sin x

f(0) = 0

First dervative:

f '(x) = cos x

f '(0) = cos 0 = 1

Second dervative:

f ''(x) = -sin x

f ''(0) = -sin 0 = 0

Third dervative:

f '''(x) = -cos x

f '''(0) = -cos 0 = -1

Fourth dervative:

f ^iv(x) = sin x

f ^iv(0) = sin 0 = 0

We observe that this pattern will continue forever.

Now to substitute the values of these derivatives into the Maclaurin Series:

We have:

This gives us: