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Fungrim entry: 037a6e

Im ⁣(sin ⁣(x+iy))=cos ⁣(x)sinh ⁣(y)\operatorname{Im}\!\left(\sin\!\left(x + i y\right)\right) = \cos\!\left(x\right) \sinh\!\left(y\right)
Assumptions:xRandyRx \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R}
\operatorname{Im}\!\left(\sin\!\left(x + i y\right)\right) = \cos\!\left(x\right) \sinh\!\left(y\right)

x \in \mathbb{R} \,\mathbin{\operatorname{and}}\, y \in \mathbb{R}
Fungrim symbol Notation Short description
ImIm ⁣(z)\operatorname{Im}\!\left(z\right) Imaginary part
Sinsin ⁣(z)\sin\!\left(z\right) Sine
ConstIii Imaginary unit
RRR\mathbb{R} Real numbers
Source code for this entry:
    Formula(Equal(Im(Sin(Add(x, Mul(ConstI, y)))), Mul(Cos(x), Sinh(y)))),
    Variables(x, y),
    Assumptions(And(Element(x, RR), Element(y, RR))))

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

2019-08-21 11:44:15.926409 UTC