sinh^-1(x)=?是否=1/sinh(x)?

Question

sinh^-1 (x)=?是否=1/sinh(x)?

Answer 1

虽然
Sin^(-1)(x)=1/Sin(x)
但Sinh^(-1)(x)并不等于1/Sinh(x)
答案是sinh^(-1)(x) = ln(x + √(x^2 + 1))
【求值过程】
sinh^(-1)(x) = y
x = sinh(y)
x = (e^y - e^(-y))／2
- 2x = - (e^y - e^(-y))
e^y - 2x - e^(-y) = 0
(e^(-y)) ((e^y)^2 - 2x (e^y) - 1) = 0
由于e^(-y) > 0,则
(e^y)^2 - 2x (e^y) - 1 = 0
(e^y - x)^2 - x^2 - 1 = 0
(e^y - x)^2 = x^2 + 1
e^y - x = ±√(x^2 + 1)
e^y = x±√(x^2 + 1)
由于e^y > 0 ,x - √(x^2 + 1) < 0
则e^y ≠ x - √(x^2 + 1)
e^y = x + √(x^2 + 1)
y = ln(x + √(x^2 + 1))
sinh^(-1)(x) = ln(x + √(x^2 + 1))
供参考