I Small angle expansions for sin, cos, and tan

1. Jul 21, 2017

Mr Davis 97

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?

2. Jul 21, 2017

andrewkirk

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.

3. Jul 21, 2017

Mr Davis 97

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?

4. Jul 21, 2017

Yes

5. Aug 1, 2017

vanhees71

It's better to write something like
$$\sin x=x+\mathcal{O}(x^3).$$
Then it's clear that the next term in the expansion is at order $x^3$.