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# From the formula, k(x)=2sin(x+pi)/(sin(x+pi)-3), what is k^-1(x)?

Also what is k^-1(k(x))=x?

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- Scarlet ManukaLv 77 years agoFavorite Answer
k(x) = 2 sin (x+π) / (sin (x+π) - 3)

= 2 (- sin x) / (- sin x - 3)

= 2 sin x / (sin x + 3)

= [2 (sin x + 3) - 6] / (sin x + 3)

= 2 - 6 / (sin x + 3)

So for k^-1(x) we get

x = 2 - 6 / (sin k^-1(x) + 3)

=> 6 / (sin k^-1(x) + 3) = 2 - x

=> sin k^-1(x) + 3 = 6 / (2 - x)

=> sin k^-1(x) = 6 / (2 - x) - 3 = (6 - 6 + 3x) / (2 - x) = 3x / (2-x)

=> k^-1(x) = arcsin (3x / (2-x)).

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