Lagrange's Trigonometric Identities/Sine

Lagrange's Sine Identity

Cosine Form of Lagrange's Sine Identity

$\ds \sum_{k \mathop = 0}^n \sin k \theta = \dfrac {\map \cos {\frac 1 2 \theta} - \map \cos {n \theta + \frac 1 2 \theta} } {2 \map \sin {\frac 1 2 \theta} }$

Sine Form of Lagrange's Sine Identity

\(\ds \sum_{k \mathop = 0}^n \sin k x\)

\(=\)

\(\ds \sin 0 + \sin x + \sin 2 x + \sin 3 x + \cdots + \sin n x\)

\(\ds \)

\(=\)

\(\ds \frac {\sin \frac {\paren {n + 1} x} 2 \sin \frac {n x} 2} {\sin \frac x 2}\)

where $x$ is not an integer multiple of $2 \pi$.

Also see

• Lagrange's Cosine Identity

Source of Name

This entry was named for Joseph Louis Lagrange.