Sine to Power of Odd Integer
Jump to navigation
Jump to search
Theorem
\(\ds \sin^{2 n + 1} \theta\) | \(=\) | \(\ds \frac {\paren {-1}^n} {2^{2 n} } \sum_{k \mathop = 0}^n \paren {-1}^k \binom {2 n + 1} k \sin \paren {2 n - 2 k + 1} \theta\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {-1}^n} {2^{2 n} } \paren {\map \sin {2 n + 1} \theta - \binom {2 n + 1} 1 \map \sin {2 n - 1} \theta + \cdots + \paren {-1}^n \binom {2 n + 1} n \sin \theta}\) |
Proof
\(\ds \sin^{2 n + 1} \theta\) | \(=\) | \(\ds \paren {\frac {e^{i \theta} - e^{-i \theta} } {2 i} }^{2 n + 1}\) | Euler's Sine Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {e^{i \theta} - e^{-i \theta} }^{2 n + 1} } {\paren {2 i}^{2 n + 1} }\) | Power of Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {e^{i \theta} - e^{-i \theta} }^{2 n + 1} } {\paren {2 i} \paren 2^{2 n} \paren {i}^{2 n} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {-1}^n} {2^{2 n} 2 i} \paren {e^{i \theta} - e^{-i \theta} }^{2 n + 1}\) | $i^2 = -1$, Power of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {-1}^n} {2^{2 n} 2 i} \sum^{2 n + 1}_{k \mathop = 0} \binom {2 n + 1} k e^{\paren {2 n + 1 - k } i \theta} \paren {-1}^k e^{-\paren {k} i \theta}\) | Binomial Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {-1}^n} {2^{2 n} 2 i} \sum^{2 n + 1}_{k \mathop = 0} \paren {-1}^{k} \binom {2 n + 1} k e^{\paren {2 n + 1 - 2 k} i \theta}\) | Exponential of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {-1}^n} {2^{2 n} 2 i} \paren {\sum^n_{k \mathop = 0} \paren {-1}^{k} \binom {2 n + 1} k e^{\paren {2 n + 1 - 2 k} i \theta} + \sum^{2 n + 1}_{k \mathop = n + 1} \paren {-1}^{k} \binom {2 n + 1} k e^{\paren {2 n + 1 - 2 k} i \theta} }\) | partitioning the sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {-1}^n} {2^{2 n} 2 i} \paren {\sum^n_{k \mathop = 0} \paren {-1}^k \binom {2 n + 1} k e^{\paren {2 n - 2 k + 1} i \theta} + \sum^n_{k \mathop = 0} \paren {-1}^{2 n + 1 - k} \binom {2 n + 1} {2 n + 1 - k} e^{\paren {2 n + 1 - 2 \paren {2 n + 1 - k} } i \theta} }\) | $k \mapsto 2 n + 1 - k$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {-1}^n} {2^{2 n} 2 i} \paren {\sum^n_{k \mathop = 0} \paren {-1}^k \binom {2 n + 1} k e^{\paren {2 n - 2 k + 1} i \theta} + \sum^n_{k \mathop = 0} \paren {-1}^{2 n} \paren {-1}^1 \paren {-1}^{- k} \binom {2 n + 1} {2 n + 1 - k} e^{\paren {2 n + 1 - 2 \paren {2 n + 1 - k} } i \theta} }\) | Product of Powers | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {-1}^n} {2^{2 n} 2 i} \paren {\sum^n_{k \mathop = 0} \paren {-1}^k \binom {2 n + 1} k e^{\paren {2 n - 2 k + 1} i \theta} - \sum^n_{k \mathop = 0} \paren {-1}^k \binom {2 n + 1} k e^{-\paren {2 n - 2 k + 1} i \theta} }\) | Symmetry Rule for Binomial Coefficients | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {-1}^n} {2^{2 n} } \paren {\sum^n_{k \mathop = 0} \paren {-1}^k \binom {2 n + 1} k \frac {e^{\paren {2 n - 2 k + 1} i \theta} - e^{-\paren {2 n - 2 k + 1} i \theta} } {2 i} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {-1}^n} {2^{2 n} } \sum_{k \mathop = 0}^n \paren {-1}^k \binom {2 n + 1} k \sin \paren {2 n - 2 k + 1} \theta\) | Euler's Sine Identity |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.70$
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Miscellaneous Problems: $130 \ \text{(b)}$