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.
Man, I've been on here for a little while and I've been making soooo many mistakes... but I guess mistakes is where I'll learn from...
By the way, there's a geometry question in the other threads... which I'm still stumped on, but it's 3am, I think I'll head to bed, and either hope someone else has the solution or I'll work it out tomorrow.
G'night, TPH (your intials)