Small Angle approximation

Just a note on the small angle approximation!

This approximation is that \(\sin\theta\approx \theta\) when \(\theta\) is in RADIANS.   The angle must be in radians for this to work.  For instance, I cannot say that \(\sin 0.1^{\circ}\approx 0.1\).  If I want the sine of the small angle \(0.1^{\circ}\), I need to first convert it to radians.  There are 57.14 degrees in one radian. Another number to know is that there are 3600 arcseconds in one degree---also, 3600 seconds in one hour.  So I like to remember that degrees are to arcesconds as hours are to seconds.

Finally, a good number to know is that 1 radian  \(=2\times 10^5\) arcseconds.

Thus, if we have an object that has angular size 2 arcseconds, it is \(10^{-5}\) radians.

And, for those of you who haven't seen the small angle approximation before, it just comes from Taylor series.

We have

$$\sin\theta\approx 0+\theta^2+\cdots$$ for small \(\theta\), and
$$\cos\theta\approx 1+\cdots$$ for small \(\theta\)  just by using Taylor's theorem (Taylor series).

Note that this means
$$\tan\theta\approx \sin\theta\approx\theta$$ for small angles (in RADIANS!).