(For the sake of intuition, 1/√0=0)
I always just used my left hand with the thumb and pinky as the y and x axes respectively, then the in-between fingers just help me visualize which "point" is higher (we only have two options to choose from if they're not equal anyways)
My trick for this was:
0 30 45 60 90 deg
---------------------------
0 1 2 3 4
Sine function of the top equals half the square root of the bottom number. Cosine is just in reverse order.
I'm not sure I understand your mnemonic. It looks like you've got 1/sqrt(3) as one of your exact value for sine and cosine... but that surely isn't right.
I always found these mnemonics just as hard to remember as the actual values so I just draw an isosceles right triangle and an equilateral one. It gives you the values for 30, 45 and 60 degrees.