How do you evaluate #sin^-1(sin((5pi)/6))#?

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Answer 1 · 2016-08-25T10:43:31

#5/6pi#.

The calculator is programmed to give only the principal value for

inverse trigonometric functions.

#5/6pi# radian = #150^o and sin 150^o# is displayed as 0.5#,

and inversely, #arc sin (0.5)# is displayed as the principal value

#30^o=pi/6# radian.

Considering sin x as a bijective function over a short interval

enclosing #x = 5/6pi#, the answer is #5/6pi#, using that

#f^(-1)f(a) = a#.

If the answer is sought as the principal value in #(-pi/2, pi/2), it is

#pi/6# that is yet another solution from the set of general values

#npi+(-1)^n(5/6pi), n = 0, +-1, +-2, +-3, ..#, for #n = -1#..