Degrees are a real PITA when it comes to doing anything even closely related to solving maths and physics problems with trigonometric functions.
Degrees might be nice for building and drawing stuff where measurements yield an easy to remember number (though pi might be the largest number a few people
do remember, so there'd be nothing against using radians for everyday purposes, were it not for the relative scarcity of said individuals), but as soon as you have equations to solve, radians are the way to go.
Some people go even further by introducing Complex Numbers and writing trigonometric functions as exponential functions in the complex plane, thereby simplifying the process of solving equations even further.
So there's my advice: Start using radians
now. There are, like, 4 values to learn for each trigonometric functions for this to be useful.
If you want to be hardcore, don't bother using radians or even introducing sine and cosine and go for the complex notation instead. This will spare you the trouble of actually learning
ANYTHING about these trigonometric functions, as values at certain points will simply appear to you by looking at the formula:
\sin(\phi) = \frac{\mathrm{e}^{i\phi} - \mathrm{e}^{-i\phi}}{2i}
\cos(\phi) = \frac{\mathrm{e}^{i\phi} + \mathrm{e}^{-i\phi}}{2}
Let's suppose you didn't know jack shit except addition, multiplication, powers, fractions and numbers on the complex plane in general (the latter is not required, however) and wanted to know the values of
\sin(\phi) and
\cos(\phi) for
\phi = 0.
By closely looking at the above equations, we see
\mathrm{e}^{i\phi} = 1 and
\mathrm{e}^{-i \phi} = 1. The sine seems to be 0/2i, which is clearly 0, whereas the cosine is 2/2, which equals 1.
There are countless other examples of why using radians and especially the complex notation greatly simplifies math, the universe and life in general.
This posting was brought to you by the Making A Better World Through Conducting Life On The Complex Plane Foundation