Lagrange's Trigonometric Identities
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Theorem
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 $2 \pi$.
Lagrange's Cosine Identity
| \(\ds \frac 1 2 + \sum_{k \mathop = 1}^n \map \cos {k x}\) | \(=\) | \(\ds \frac 1 2 + \cos x + \cos 2 x + \cos 3 x + \cdots + \cos n x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\map \sin {\paren {2 n + 1} x / 2} } {2 \map \sin {x / 2} }\) |
where $x$ is not an integer multiple of $2 \pi$.
Also known as
Lagrange's Trigonometric Identities are also known as:
The identities version seems more common.
Source of Name
This entry was named for Joseph Louis Lagrange.