Question

Explain why sin^-1[sin(3pi/4)] does not = 3pi/4 when y=sin(x) and y=sin^-1(x) are inverses.

Any help on this question is greatly appreciated. Thank you!

Answer 1

sin(3PI/4) is a specific number. The inverse of that number can track to several angles, all of which have the same value of sine.

Example; arcsin(sin30deg).
well sin 30 deg=.5
but arcsin(.5) can be 30 deg, 150 deg, and so on...

Answer 2

well sin 3 pi/4 = +1/sqrt 2 = +sqrt 2/ 2 in quadrant 2
BUT
sin pi/4 = +sqrt2 / 2 in quadrant 1
sin(3pi/4) = sin(pi/4)
sin^-1 (sqrt2 /2) = 45 degrees and sin^-1(sqrt2/2) = 135 degrees
so for example
sin^-1 (sin3pi/4) = pi/4