Do Sine and Arcsine Cancel Each Other Out?

Yes, sine and arcsine are inverse functions of each other, and they can indeed cancel each other out under specific conditions.

For instance, if you take the sine of an angle and then apply arcsine to that result, you will get back the original angle, as long as the angle falls within the correct range. Mathematically, this can be expressed as:

arcsin(sin(x)) = x  for -π/2 ≤ x ≤ π/2

Similarly, on the other hand, if you take arcsine of a value that is within the range of -1 to 1 (which are the possible outputs for the sine function), and then apply the sine function to it, you will get back the original value:

sin(arcsin(y)) = y  for -1 ≤ y ≤ 1

Therefore, while sine and arcsine do ‘cancel’ each other out, it’s important to remember the limitations on the input values to ensure the operations are valid. If you step outside these ranges, then the functions will not revert to the original input.