Compute $\sin 1755^\circ$.
I know that the sine function is periodic with a period of 360 degrees, so I can reduce the angle 1755 degrees by subtracting multiples of 360 until I get an angle between 0 and 360 degrees. I can divide 1755 by 360 to see how many multiples of 360 there are. I get 1755 / 360 = 4.875, so there are 4 full multiples of 360 and a remainder of 0.875. To get the remainder in degrees, I multiply 0.875 by 360 and get 315. So $\sin 1755^\circ = \sin 315^\circ$. Now I can use the reference angle of 45 degrees and the fact that sine is negative in the fourth quadrant to get the answer. I have $\sin 315^\circ = -\sin 45^\circ = -\frac{\sqrt{2}}{2}$. 