I'm back problem (Double Euler's?)

Algebra Level 5

\[\large \ln(k) = \exp\left[ -\ln(3) + i \sin^{-1} \left( \frac{24}{25} \right) \right]\]

Suppose we define \(k\) such that the equation above is fulfilled, and if the imaginary part of \(k\) can be expressed in the form of \[\large e^{\frac AB} \sin\left( \frac CD\right) \]

where \(A,B,C\) and \(D\) are positive integers such that \( \gcd(A,B) = \gcd(C,D) = 1\), find the value of \(A+B+C+D\).

Details and Assumptions

We define \(\exp (x) \) as \(e^x\).


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