1. ## Polynomial approximation

Here's a challenge: for the function $y= a\,\sin{(bx+c)} + d$, find a polynomial of degree 3 (ie a cubic) approximation for y on the interval from one trough to one peak (about the main original point of inflection) - eg for $y=\sin{(x)}$ that interval would be $\left[-\frac{\pi}{2}, \,\frac{\pi}{2}\right]$

Express the answer in the form $y = px^3+qx^2+rx+s$, where p, q , r and s are in terms of a, b, c and d.

2. It doesnt seem to work when $c \neq 0$ I have no idea why?

3. Originally Posted by nzmathman
Here's a challenge: for the function $y= a\,\sin{(bx+c)} + d$, find a polynomial of degree 3 (ie a cubic) approximation for y on the interval from one trough to one peak (about the main original point of inflection) - eg for $y=\sin{(x)}$ that interval would be $\left[-\frac{\pi}{2}, \,\frac{\pi}{2}\right]$

Express the answer in the form $y = px^3+qx^2+rx+s$, where p, q , r and s are in terms of a, b, c and d.
What is the norm against which the quality of the fit between the polynomial and function is to be judged?

CB

4. Hi, basically you find the approximation using certain points of the sine curve so that there is exactly enough to solve for a, b, c and d. (this is giving it away a bit but I used the point of inflection and one of the stationary points). Never mind my earlier post about it not working when c was not zero, I made a mistake in the formula but fixed it now