I was quite satisfied with the approach my instructors took with me.

In the first semester course in calculus, we covered some properties about conics, polar coordinates, and don`t go past differentiation, where $\sin(x)$ can be differentiated with the limit definition, independent of how you first define sine. Thus we kept our approach at the high school level, relating the sine and cosine functions to the unit circle.

In second semester calculus, one starts learning about power series, and it is now natural to present the sine and cosine functions in their power series representations, where with the addition of a little complex analysis (also introduced in second semester calculus) can be related to the power series for $e^x$ and indulge Euler's identity.

The differential equation version was never really emphasized, though certainly mentioned after you first find the derivatives of sine and cosine and (hopefully) notice a pattern right away. (I believe this was something along the lines of an assignment problem asking for the $n$-th derivative of $sin(x)$, where $n$ was quite large.)

I suppose this doesn't really answer the question though.. If I were teaching a calculus course to student who had already seen a construction of $\mathbb{R}$, I would go with the power series definitions. The unit circle should really be review from high school, Euler's identity should really focus more on the connection of the functions instead of being a first glance, and the differential equation, well, you can see this from the power series representation anyway!

I was quite satisfied with the approach my instructors took with me.

In the first semester course in calculus, we covered some properties about conics, polar coordinates, and don't go past differentiation, where $\sin(x)$ can be differentiated with the limit definition, independent of how you first define sine. Thus we kept our approach at the high school level, relating the sine and cosine functions to the unit circle.

In second semester calculus, one starts learning about power series, and it is now natural to present the sine and cosine functions in their power series representations, where with the addition of a little complex analysis (also introduced in second semester calculus) can be related to the power series for $e^x$ and indulge Euler's identity.

The differential equation version was never really emphasized, though certainly mentioned after you first find the derivatives of sine and cosine and (hopefully) notice a pattern right away. (I believe this was something along the lines of an assignment problem asking for the $n$-th derivative of $\sin(x)$, where $n$ was quite large.)

I suppose this doesn't really answer the question though.. If I were teaching a calculus course to student who had already seen a construction of $\mathbb{R}$, I would go with the power series definitions. The unit circle should really be review from high school, Euler's identity should really focus more on the connection of the functions instead of being a first glance, and the differential equation, well, you can see this from the power series representation anyway!