Category:Tangent of Sum of Series of Angles
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This category contains pages concerning Tangent of Sum of Series of Angles:
Let $\theta_1, \theta_2, \ldots, \theta_n$ be angles.
For all $k \in \set {1, 2, 3, \ldots, n}$, let $s_k$ be defined as the sum of the product of $\map \tan {\theta_1}, \map \tan {\theta_2}, \ldots, \map \tan {\theta_n}$ taken $k$ at a time:
| \(\ds s_1\) | \(=\) | \(\ds \sum_{i \mathop = 1}^n \map \tan {\theta_i}\) | ||||||||||||
| \(\ds s_2\) | \(=\) | \(\ds \sum_{i \mathop = 1}^{n - 1} \sum_{j \mathop = i + 1}^n \map \tan {\theta_i} \map \tan {\theta_j}\) | ||||||||||||
| \(\ds s_3\) | \(=\) | \(\ds \sum_{i \mathop = 1}^{n - 2} \sum_{j \mathop = i + 1}^{n - 1} \sum_{k \mathop = j + 1}^n \map \tan {\theta_i} \map \tan {\theta_j} \map \tan {\theta_k}\) | ||||||||||||
| \(\ds \) | \(\vdots\) | \(\ds \) | ||||||||||||
| \(\ds s_k\) | \(=\) | \(\ds \sum_{\substack {S \mathop \in \set {1, 2, \ldots, n} \\ \size S \mathop = k} } \paren {\prod_{j \mathop \in S} \tan \theta_j}\) |
Then:
- $\map \tan {\theta_1 + \theta_2 + \theta_3 + \cdots + \theta_n} = \dfrac {s_1 - s_3 + s_5 - \cdots} {1 - s_2 + s_4 - \cdots}$
Pages in category "Tangent of Sum of Series of Angles"
The following 3 pages are in this category, out of 3 total.