I apologize for the confusion. Let me explain step by step how to get the expression for arcsin(4) in the form a+ib.
1. Start with the equation: 4 = sin(a)cosh(b) + i cos(a)sinh(b)
2. To simplify, we need to take the inverse hyperbolic cosine (arccosh) of both sides. This will isolate the b term.
3. Apply the arccosh function to both sides of the equation. The arccosh function is the inverse of the hyperbolic cosine function.
4. By applying arccosh to both sides, we get: arccosh(4) = arccosh(sin(a)cosh(b) + i cos(a)sinh(b))
5. Now, to determine the value of a, we can look at the real part of the equation. The real part is sin(a)cosh(b). If we equate this to zero, we can solve for a.
6. Setting sin(a)cosh(b) = 0 gives us sin(a) = 0. This means that a must be pi/2 (90 degrees) or any integer multiple of pi/2, as sin(a) = 0 for those values of a.
7. With a = pi/2, the equation simplifies to: arccosh(4) = arccosh(cosh(b))
8. Now, we can solve for b by isolating it on one side. This gives us b = arccosh(4).
So, in summary, arcsin(4) in the form a+ib is a = pi/2 and b = arccosh(4).