There are no simple way of doing it I don't think...
There's other ways I can think of...
a) Using a sine table, not as common in Australia and in modern times where calculators are so convenient.
b) sin (105.5) = sin (90 + 15.5)... use your sine expansion, then expand sin (15.5) using sin (15 + 0.5), sin (15) is easy enough to do, just use sin (45 - 30), as for the sin (0.5), you can either remember its value, or just say it's small enough for a hand calculation to ignore... cos it only effect the answer by 0.0087, so if you give your answer to 1 decimal place, this won't affect it at all.
c) Find a sine graph, and read the value?
d) May be double angle formulae will come in handy?
e) That, or you could try the Series expansion for a Sine function: see
Trigonometric function - Wikipedia, the free encyclopedia for more details... It might be easier to carry out step b), then use this for sin (0.5)
These are all just suggestions, and it only gives an approximately correct answer.