How the small angle approximation can be used

In class, we discussed how the small angle approximation can be used to simplify a lot of math in astronomy. This approximation says that for “small” angles that are expressed in radians, we can say sinθ≈tanθ≈θ.

a) (2 pts) Assume θ=0.00012345 radians. Directly calculate sin(θ) and tan(θ), quoting the values you get to 5 significant digits. Discuss whether you think the small angle approximation does an OK job in this situation.

b) (1 pt) Look at this plot, which shows θ, sin(θ), and tan(θ) as a function of θ (with all angle units in radians). Based on these curves, up to how large of an angle does the small angle approximation still, does a pretty good job?

c) (3 pts) Astronomers commonly use units of “arcseconds” or “arcminutes” to measure small angles in the sky. These are defined so that 60 arcseconds=1 arcminute and 60 arcminutes=1˚. Calculate sin⁡(1 arcsecond). State whether or not the small angle approximation is accurate for this angle.