#3 sin(x) = -sqrt(2)/2 so x = 315 degrees. So we have sin(x+pi/3) = sin(375) = sin(15) which in exact form would be (sqrt(3) -1)/(2sqrt(2))
#4 to find x we take the arctan(20/21) which is approximately 43.6 degrees and add 180 to it since it's in the 3rd quadrant which is 223.6 degrees so we do sin(223.6 + 90) = sin(313.6) which equals -21/29. This can also be done with the pythagorean theorem. -21/29.
#5 13^2 - 5^2 = 12^2. so sin is -12/-13 = 12/13. we take arcsin of that and add 180 and then divide by 2 and take the sin of that and its approximately 3/sqrt(13).
#6 sin is 4/5 so we take arcsin of that and get 143.13 which -60 is 83.13 and we take the sin of that to get 3/10 + 2sqrt(3)/5
Hope this helped :L