Small angle expansions for sin, cos, and tan

[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?

 

[andrewkirk]

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?

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]

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.

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]

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?

Yes

 

[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 $x^3$.