Calculate sin(100°) without a calculator.

>So by heart, I know the way to solve this would be this...

>sin(100°) = sin(70° + 30°)

Why's that?  You don't know sin(70°) either.

>But how on earth does sin(70° + 30°) = sin(70°)*cos(30°) + sin(30°)*cos(70°) ???

>And then there's cos(100°) which will be = cos(70°)*cos(30°) - sin(70°)*sin(30°)

These are trig identities that you should definitely know.  I mean, where did you even get these from if you don't know them?  How did you write these lines down?
I bet there's a nice geometric way to do it, but I prefer to just use complex numbers.

One thing you should know is Euler's identity: e^iø = cos(ø) + i·sin(ø).  Then, e^i·(a+b) = cos(a + b) + i·sin(a + b), right?  But e^i·(a+b) = e^ia · e^ib = (cos(a) + i·sin(a)) · (cos(b) + i·sin(b)).  Expand those out to get cos(a + b) + i·sin(a + b) = cos(a)cos(b) + i·cos(a)sin(b) + i·sin(a)cos(b) – sin(a)sin(b).  If you collect the real terms, you get cos(a + b) = cos(a)cos(b) – sin(a)sin(b), and if you collect the imaginary terms, you get sin(a + b) = cos(a)sin(b) + sin(a)cos(b) (or in the other order, but it's the same thing).  That's pretty simple, no?