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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(x) \sinh(y)
Assumptions:xR  and  yRx \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R}
\operatorname{Im}\!\left(\sin\!\left(x + i y\right)\right) = \cos(x) \sinh(y)

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R}
Fungrim symbol Notation Short description
ImIm(z)\operatorname{Im}(z) Imaginary part
Sinsin(z)\sin(z) Sine
ConstIii Imaginary unit
Coscos(z)\cos(z) Cosine
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.

2021-03-15 19:12:00.328586 UTC