Equations with transformed arguments

Set the whole argument equal to a new angle and transform the interval before solving. Example: sin(2x-30)= 12, 0 <= x 360 theta=2x-30 -30 <= theta 690 **Step** | **Working** Reference angle | alpha=sin-1 12=30 Quadrants I and II (sine positive) | 2x-30 =30, 150 Add 360 within -30 <= 2x-30 <= 690 | 2x-30 =30, 150, 390, 510 Return to x | 2x=60, 180, 420, 540 x=30, 90, 210, 270 Add full periods inside the transformed interval, then recover x and check the original bounds. A factor of 2 in the argument doubles the number of cycles, but endpoint solutions still need checking.