Sine to Power of Odd Integer

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Theorem

\(\displaystyle \sin^{2 n + 1} \theta\) \(=\) \(\displaystyle \frac {\paren {-1}^n} {2^{2 n} } \sum_{k \mathop = 0}^n \paren {-1}^k \binom {2 n + 1} k \sin \paren {2 n - 2 k + 1} \theta\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {2^{2 n} } \paren {\map \sin {2 n + 1} \theta - \binom {2 n + 1} 1 \, \map \sin {2 n - 1} \theta + \cdots + \paren {-1}^n \binom {2 n + 1} n \sin \theta}\) $\quad$ $\quad$


Proof

\(\displaystyle \sin^{2 n + 1} \theta\) \(=\) \(\displaystyle \paren {\frac {e^{i \theta} - e^{-i \theta} } {2 i} }^{2 n + 1}\) $\quad$ Sine Exponential Formulation $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {e^{i \theta} - e^{-i \theta} }^{2 n + 1} } {\paren {2 i}^{2 n + 1} }\) $\quad$ Exponent Combination Laws: Power of Product $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {i \paren {-1}^n} {2^{2 n + 1} } \paren {e^{-i \theta} - e^{i \theta} }^{2 n + 1}\) $\quad$ Exponent Combination Laws: Power of Power $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {i \paren {-1}^n} {2^{2 n + 1} } \sum^{2 n + 1}_{k \mathop = 0} \binom {2 n + 1} k e^{-k i \theta} \paren {-1}^{2 n - k + 1} e^{\paren {2 n - k + 1} i \theta}\) $\quad$ Binomial Theorem, Exponent Combination Laws: Power of Product $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {i \paren {-1}^n} {2^{2 n + 1} } \sum^{2 n + 1}_{k \mathop = 0} \binom {2 n + 1} k \paren {-1}^{2 n - k + 1} e^{\paren {2 n - 2 k + 1} i \theta}\) $\quad$ Exponent Combination Laws: Product of Powers $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {i \paren {-1}^n} {2^{2 n + 1} } \paren {\sum^n_{k \mathop = 0} \binom {2 n + 1} k \paren {-1}^{2 n - k + 1} e^{\paren {2 n - 2 k + 1} i \theta} + \sum^{2 n + 1}_{k \mathop = n + 1} \binom {2 n + 1} k \paren {-1}^{2 n - k + 1} e^{\paren {2 n - 2 k + 1} i \theta} }\) $\quad$ Partitioning the sum $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {i \paren {-1}^n} {2^{2 n + 1} } \paren {\sum^n_{k \mathop = 0} \binom {2 n + 1} k \paren {-1}^{2 n - k + 1} e^{\paren {2 n - 2 k + 1} i \theta} + \sum^n_{k \mathop = 0} \binom {2 n + 1} {2 n + 1 - k} \paren {-1}^k e^{\paren {2 n - 2 paren {2 n + 1 - k} + 1} i \theta} }\) $\quad$ $k \mapsto 2n+1-k$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {i \paren {-1}^n} {2^{2 n + 1} } \paren {\sum^n_{k \mathop = 0} \binom {2 n + 1} k \paren {-1}^{2 n - k + 1} e^{\paren {2 n - 2 k + 1} i \theta} + \sum^n_{k \mathop = 0} \binom {2 n + 1} k \paren {-1}^k e^{-\paren {2 n - 2 k + 1} i \theta} }\) $\quad$ Symmetry Rule for Binomial Coefficients $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {i \paren {-1}^n} {2^{2 n + 1} } \paren {-\sum^n_{k \mathop = 0} \binom {2 n + 1} k \paren {-1}^k e^{\paren {2 n - 2 k + 1} i \theta} + \sum^n_{k \mathop = 0} \binom {2 n + 1} k \paren {-1}^k e^{-\paren {2 n - 2 k + 1} i \theta} }\) $\quad$ Exponent Combination Laws: Product of Powers, $\paren {-1}^{2 n} = 0$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {-1}^n } {2^{2 n} } \sum^n_{k \mathop = 0} \binom {2 n + 1} k \map \sin {2 n - 2 k + 1} \theta\) $\quad$ Sine Exponential Formulation $\quad$

$\blacksquare$



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