# Fungrim entry: 3fb3ca

$\sin\!\left(z\right) = \frac{2 \tan\!\left(\frac{z}{2}\right)}{\tan^{2}\!\left(\frac{z}{2}\right) + 1}$
Assumptions:$z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \notin \left\{ \left(2 n + 1\right) \pi : n \in \mathbb{Z} \right\}$
Alternative assumptions:$z \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, z \notin \left\{ \left(2 n + 1\right) \pi : n \in \mathbb{Z} \right\}$
TeX:
\sin\!\left(z\right) = \frac{2 \tan\!\left(\frac{z}{2}\right)}{\tan^{2}\!\left(\frac{z}{2}\right) + 1}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \notin \left\{ \left(2 n + 1\right) \pi : n \in \mathbb{Z} \right\}

z \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, z \notin \left\{ \left(2 n + 1\right) \pi : n \in \mathbb{Z} \right\}
Definitions:
Fungrim symbol Notation Short description
Sin$\sin\!\left(z\right)$ Sine
Pow${a}^{b}$ Power
CC$\mathbb{C}$ Complex numbers
SetBuilder$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$ Set comprehension
ConstPi$\pi$ The constant pi (3.14...)
ZZ$\mathbb{Z}$ Integers
FormalPowerSeries$K[[x]]$ Formal power series
Source code for this entry:
Entry(ID("3fb3ca"),
Formula(Equal(Sin(z), Div(Mul(2, Tan(Div(z, 2))), Add(Pow(Tan(Div(z, 2)), 2), 1)))),
Variables(z),
Assumptions(And(Element(z, CC), NotElement(z, SetBuilder(Mul(Add(Mul(2, n), 1), ConstPi), n, Element(n, ZZ)))), And(Element(z, FormalPowerSeries(CC, x)), NotElement(z, SetBuilder(Mul(Add(Mul(2, n), 1), ConstPi), n, Element(n, ZZ))))))

## Topics using this entry

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