I have seen many times that for small angles sinθ≈θ, i want to know for how small value?

Question

and i think the value is less than 1 right? certainly sin1 ≠ 1. here units i used are degrees.

In: 6

Answer 1

>sinθ≈θ

Isn’t exactly right. It’s just what we tell younger students, so that we don’t have to teach them perturbation theory.

Really the statement should be

>sinθ~θ, for θ –> 0

Which is a shorthand for saying that sinθ/θ tends toward 1 as θ tends toward 0.

It isn’t really saying that these two things are “approximately equal”. Rather, the fact that the ratio between the two gets closer and closer to 1, indicates that they are “changing in a similar way”, so the isn’t much space for the two to differ.

EDIT:

In similar fashion your teachers will likely say “much less than” and use a double < symbol to represent it

> ” << ”

This is also a cheeky adaptation of notation from the same field of mathematics.

EDIT EDIT:

As others have said, sin needs to be in radians, but that is generally considered to be the default assumption when working with sin in any mathematical context.

EDIT EDIT EDIT:

If you are computationally minded person, you may know “big o notation”.

>sinθ~θ, for θ –> 0

is just more stringent version of:

>sinθ=O(θ), for θ –> 0

EDIT IV: Not sure why I’m getting down voted?

EDIT V: Crap I’m on explain like I’m five -_-