Fungrim entry: 037a6e

\operatorname{Im}\!\left(\sin\!\left(x + i y\right)\right) = \cos(x) \sinh(y)

Assumptions:

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R}

TeX:

\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}

Definitions:

Fungrim symbol	Notation	Short description
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Sin`	$\sin(z)$	Sine
`ConstI`	$i$	Imaginary unit
`Cos`	$\cos(z)$	Cosine
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("037a6e"),
    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))))

Topics using this entry

Sine

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