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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\!\left(a\right) \cosh\!\left(b\right) + i \cos\!\left(a\right) \sinh\!\left(b\right)
Assumptions:a∈C and⁡ b∈Ca \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C}
TeX:
\sin\!\left(a + b i\right) = \sin\!\left(a\right) \cosh\!\left(b\right) + i \cos\!\left(a\right) \sinh\!\left(b\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
Sinsin⁡ ⁣(z)\sin\!\left(z\right) Sine
ConstIii Imaginary unit
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("3b839c"),
    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.

2019-06-18 07:49:59.356594 UTC