Assignment task: 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) 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) 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) Astronomers commonly use units of "arcseconds" or "arcminutes" to measure small angles on 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.