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Fungrim entry: 3fb3ca

sin(z)=2tan ⁣(z2)tan2 ⁣(z2)+1\sin(z) = \frac{2 \tan\!\left(\frac{z}{2}\right)}{\tan^{2}\!\left(\frac{z}{2}\right) + 1}
Assumptions:zCandz{(2n+1)π:nZ}z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \notin \left\{ \left(2 n + 1\right) \pi : n \in \mathbb{Z} \right\}
Alternative assumptions:zC[[x]]andz{(2n+1)π:nZ}z \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, z \notin \left\{ \left(2 n + 1\right) \pi : n \in \mathbb{Z} \right\}
TeX:
\sin(z) = \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
Sinsin(z)\sin(z) Sine
Powab{a}^{b} Power
CCC\mathbb{C} Complex numbers
ConstPiπ\pi The constant pi (3.14...)
ZZZ\mathbb{Z} Integers
FormalPowerSeriesK[[x]]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, Set(Mul(Add(Mul(2, n), 1), ConstPi), ForElement(n, ZZ)))), And(Element(z, FormalPowerSeries(CC, x)), NotElement(z, Set(Mul(Add(Mul(2, n), 1), ConstPi), ForElement(n, ZZ))))))

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

2019-10-05 13:11:19.856591 UTC