Derive the first three (non-zero) terms of Taylor's series expansion for the function (a) f(x)=sin(x)f(x)=sin⁡(x)  about the origin and thereby estimate sin(0.2)=​

We want to find the first 3 terms of the Taylor's series expansion for f(x) = sin(x) around x = 0.

Remember that a Taylor's series expansion of a function f(x) around the point x₀ is given by:

Where in the formula we have the first 3 terms of the expansion (but there are a lot more).

So, if:

f(x) = sin(x)

x₀ = 0

The terms are:

We already can see that the next term is zero (because when we derive the cos part, we will get a sin() that is zero when evaluated in x = 0), then the next non zero term is:

Then we can write:

Evaluating this in x = 0.2, we get:

(5x-1)^2= (5x)^2-2(5x)×1+(1)^2=25x^2-10x+1