If we treat #arcsin# as multivalued then
#x = arcsin sin a #
is the same as
#sin x = sin a #
which has supplementary angle solutions
#x = a + 2pi k quad or quad x=(pi-a) + 2pi k quad # for integer #k#
#x = arcsin sin ({7pi}/3) #
#sin x = sin ({7pi}/3) #
#x = {7 pi}/3 + 2pi k or x = (pi- {7pi}/3) + 2pi k#
#x = {7 pi}/3 + 2pi k or x = - {pi}/3 + 2pi k#
#arcsin sin ({7pi}/3) = {7 pi}/3 + 2pi k or - {pi}/3 + 2pi k quad# integer #k.#
The principal value for the inverse sine is in the range #-pi/2# to #pi/2# so we take #k=-1# in the first clause and get
#text{Arc}text{sin}\sin ({7pi}/3) = pi/3 #
[Had this one sitting around in my browser; I forget to hit post.]
