Is it possible to make curvature of sine wave equal to that of a parabola?

Question

Suppose there is a symmetric parabola pointing downwards now we only consider the part above the x axis so is it possible to make curvature of sine wave equal to that part above x axis So that it coincides with that part of parabola above x axis sorry if this question makes no sense please don't close it if i was not able to make this more understandable then i will edit it

Answer 1

Locally you can make a sine wave and a parabola agree to second order. The Taylor series of the cosine (which is a shifted sine wave) is $1-\frac{x^2}{2!}+\frac {x^4}{4!}+\ldots$. The first two terms make a parabola centered at $0$ with a maximum of $1$. As long as you are close enough to $0$ that the $\frac {x^4}{4!}$ term is negligible, they will agree. The agreement is not exact. The Alpha plot below shows they match very closely out to $\pm \frac 12$, and rather well out to $\pm 1$