Safety of small angle approximations

[tmiddlet]

Today I was doing a problem (physics) and ended up with the a differential equation dT = -µ T sin(dθ) (Where µ is constant)

I wasn't sure what to do with the sin(dθ), so I used the small angle approximation of sin(dθ) = dθ. I would think this would be a perfect approximation because dθ is of infinitesimal size, but I'm not sure. I got the right answer, but I just wanted to make sure that this is a safe thing to do in general, or if I just got lucky.

(Sorry if this is the wrong forum, this is my first post here)

 

[hotvette]

Beam deflection theory uses the same approximation and is widely used in practice. It just means that in your case the function T(θ) you derive is only valid for small θ. How small? Hard to say, but the ratio of θ/sin(θ) differs from unity by about 1% for θ=0.25 radians (14 degrees) and 5% for θ=0.55 radians(32 degrees). In the end it depends on how much error you can tolerate.