The discussion revolves around the small angle approximations for the trigonometric functions sine, cosine, and tangent. Participants explore the classification of these approximations as first-order or second-order based on their Taylor series expansions and the implications of these classifications.
Participants express differing views on the classification of the approximations as first or second order, indicating that multiple competing perspectives remain unresolved.
There is an ongoing discussion about the definitions and implications of order in Taylor series expansions, with some assumptions about the nature of the approximations remaining unaddressed.
They are called second order because they include all terms of the Taylor series up to and including the term of order two, so that any discrepancy is of third order. Note that the second order terms are zero for sine and tan.Mr Davis 97 said:From the Wikipedia article https://en.wikipedia.org/wiki/Small-angle_approximation, it says that they are "second-order approximations." What makes all three second order? Shouldn't sin and tan be first-order and cos be second-order?
So for sin and tan are the second and first-order approximations the same? And for cos, are the zeroth and first-order approximations the same?andrewkirk said:They are called second order because they include all terms of the Taylor series up to and including the term of order two, so that any discrepancy is of third order. Note that the second order terms are zero for sine and tan.
YesMr Davis 97 said:So for sin and tan are the second and first-order approximations the same? And for cos, are the zeroth and first-order approximations the same?