Towards Maclaurin
Build series for the sine and cosine functions by adding one term at a time, alternately making the approximation too big then too small but getting ever closer.
Problem
(2) By considering the derivative of the function
(3) By considering the derivative of the function
(4) By considering the derivative of the function
(5) What can you say about continuing this process?
Getting Started
If you can show that the derivative of a function is always positive then the function is increasing and if you can show that the derivative is always negative then the function is decreasing.
Student Solutions
Teachers' Resources
You are asked in part (5) about generalising this method. What formulae would you expect to derive this way? This method shows a sequence of polynomial approximations to the trig. functions. You are not being asked to give a rigorous proof that the formulae hold in general; that requires a little more work.
Here is another route to the same result. It is not a solution to the problem because the problem specified another method. This should help in seeing the bigger picture more clearly.
We know $\cos x \leq 1$ for all $x$.
The function $f_1(x) = 1 - \cos x$ is positive for all x and so the integral of this function from 0 to $x$ is positive for all $x$. Hence
Integrating again, where $f_2(x) = x - \sin x$
Integrating again, where $f_3(x)= \cos x - \left(1 - {x^2\over 2}\right)$
Integrating again, where $f_4(x)= \sin x - \left(x - {x^3\over 3!}\right)$
and so on ...