Small Angle Question in relation to Astronomy

[astropi]

First of all this is NOT for class! Secondly, this seems rather simple, so perhaps I am simply overlooking the obvious. Anyway, here goes:

If some binary star is about 400 ly away, and a telescope gives a separation of 36" (0.01 degrees) for the binary, then can we use the small angle formula? Well, it certainly seems like we should be able to. However, the small angle formula for tangent is is just [ tex ]tan\theta \sim \theta[ /tex ]. If we then calculate the physical separation between the two stars we get 4 ly, which is incorrect. If instead of taking the small angle formula we just take the tangent of 0.01 we get 0.00017 and that in turn leads to the "correct" distance. My question becomes why does the small angle formula fail here? What am I missing? Thanks!

edit: sorry, I can not get the latex to work for some reason!

 

[phyzguy]

The small angle formula, i.e. theta ~= sin(theta) ~= tan(theta), only works when theta is measured in radians, not when theta is measured in degrees. If you convert 0.01 degrees into radians, theta(radians) = 0.01 * pi/180 = .00017. Then everything works.

 

[astropi]

The small angle formula, i.e. theta ~= sin(theta) ~= tan(theta), only works when theta is measured in radians, not when theta is measured in degrees. If you convert 0.01 degrees into radians, theta(radians) = 0.01 * pi/180 = .00017. Then everything works.

Yes, you're absolutely right. And, believe it or not, I realized that as soon as I posted (duh)!
Thanks for the reply though :)