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 $n = -1$ ..

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