Fungrim entry: e3f8a4

\sin\!\left(n z\right) = \sum_{k=0}^{\left\lfloor \left( n - 1 \right) / 2 \right\rfloor} {\left(-1\right)}^{k} {n \choose 2 k + 1} \cos^{n - 2 k - 1}\!\left(z\right) \sin^{2 k + 1}\!\left(z\right)

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}

TeX:

\sin\!\left(n z\right) = \sum_{k=0}^{\left\lfloor \left( n - 1 \right) / 2 \right\rfloor} {\left(-1\right)}^{k} {n \choose 2 k + 1} \cos^{n - 2 k - 1}\!\left(z\right) \sin^{2 k + 1}\!\left(z\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Sin`	$\sin(z)$	Sine
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Binomial`	${n \choose k}$	Binomial coefficient
`Cos`	$\cos(z)$	Cosine
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("e3f8a4"),
    Formula(Equal(Sin(Mul(n, z)), Sum(Mul(Mul(Mul(Pow(-1, k), Binomial(n, Add(Mul(2, k), 1))), Pow(Cos(z), Sub(Sub(n, Mul(2, k)), 1))), Pow(Sin(z), Add(Mul(2, k), 1))), For(k, 0, Floor(Div(Sub(n, 1), 2)))))),
    Variables(n, z),
    Assumptions(And(Element(z, CC), Element(n, ZZGreaterEqual(0)))))

Topics using this entry

Sine