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Fungrim entry: 3b839c

sin ⁣(a+bi)=sin(a)cosh(b)+icos(a)sinh(b)\sin\!\left(a + b i\right) = \sin(a) \cosh(b) + i \cos(a) \sinh(b)
Assumptions:aC  and  bCa \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}
\sin\!\left(a + b i\right) = \sin(a) \cosh(b) + i \cos(a) \sinh(b)

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C}
Fungrim symbol Notation Short description
Sinsin(z)\sin(z) Sine
ConstIii Imaginary unit
Coscos(z)\cos(z) Cosine
CCC\mathbb{C} Complex numbers
Source code for this entry:
    Formula(Equal(Sin(Add(a, Mul(b, ConstI))), Add(Mul(Sin(a), Cosh(b)), Mul(Mul(ConstI, Cos(a)), Sinh(b))))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC