Sine of Integer Multiple of Argument/Formulation 1
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Theorem
For $n \in \Z_{>0}$:
\(\ds \sin n \theta\) | \(=\) | \(\ds \sin \theta \paren {\paren {2 \cos \theta}^{n - 1} - \dbinom {n - 2} 1 \paren {2 \cos \theta}^{n - 3} + \dbinom {n - 3} 2 \paren {2 \cos \theta}^{n - 5} - \cdots}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {k + 1} } k \paren {2 \cos \theta}^{n - \paren {2 k + 1} } }\) |
Proof
The proof proceeds by induction.
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
- $\ds \sin n \theta = \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {k + 1} } k \paren {2 \cos \theta}^{n - \paren {2 k + 1} } }$
Basis for the Induction
$\map P 1$ is the case:
\(\ds \sin \theta\) | \(=\) | \(\ds \sin \theta\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sin \theta \binom {1 - 1 } 0 \paren {2 \cos \theta}^{1 - \paren {0 + 1} }\) | Zero Choose Zero, Definition of Integer Power |
So $\map P 1$ is seen to hold.
$\map P 2$ is the case:
\(\ds \sin 2 \theta\) | \(=\) | \(\ds 2 \sin \theta \cos \theta\) | Double Angle Formula for Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \sin \theta \binom {2 - 1} 0 \paren {2 \cos \theta}^{2 - \paren {0 + 1} }\) |
So $\map P 2$ is also seen to hold.
This is our basis for the induction.
Induction Hypothesis
Now we need to show that, if $\map P n$ is true, where $n > 2$, then it logically follows that $\map P {n + 1}$ is true.
So this is our induction hypothesis:
- $\ds \map \sin {n \theta} = \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {k + 1} } k \paren {2 \cos \theta}^{n - \paren {2 k + 1} } }$
from which we are to show:
- $\ds \map \sin {\paren {n + 1} \theta} = \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n + 1 - \paren {k + 1} } k \paren {2 \cos \theta}^{n + 1 - \paren {2 k + 1} } }$
Induction Step
This is our induction step:
To proceed, we will require the following lemma:
Lemma
- For $n \in \Z$:
\(\ds \map \cos {n \theta} \map \sin {\theta}\) | \(=\) | \(\ds \map \sin {n \theta} \map \cos {\theta} - \map \sin {\paren {n - 1 } \theta}\) |
$\Box$
\(\ds \map \sin {\paren {n + 1} \theta}\) | \(=\) | \(\ds \map \sin {n \theta + \theta }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sin n \theta \cos \theta + \cos n \theta \sin \theta\) | Sine of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \sin n \theta \cos \theta + \map \sin {n \theta} \map \cos {\theta} - \map \sin {\paren {n - 1 } \theta}\) | Lemma above | |||||||||||
\(\ds \) | \(=\) | \(\ds \sin n \theta \paren {2 \cos \theta } - \map \sin {\paren {n - 1 } \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {k + 1} } k \paren {2 \cos \theta}^{n - \paren {2 k + 1} } } \paren {2 \cos \theta } - \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - 1 - \paren {k + 1} } k \paren {2 \cos \theta}^{n - 1 - \paren {2 k + 1} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {k + 1} } k \paren {2 \cos \theta}^{n - \paren {2 k} } } - \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {k + 2} } k \paren {2 \cos \theta}^{n - \paren {2 k + 2} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sin \theta \paren {\paren {2 \cos \theta}^n + \paren {\sum_{k \mathop \ge 1} \paren {-1}^k \binom {n - \paren {k + 1} } k \paren {2 \cos \theta}^{n - \paren {2 k} } } + \paren {\sum_{k \mathop \ge 1} \paren {-1}^k \binom {n - \paren {k - 1 + 2} } {k - 1 } \paren {2 \cos \theta}^{n - \paren {2 \paren {k - 1 } + 2} } } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sin \theta \paren {\paren {2 \cos \theta}^n + \paren {\sum_{k \mathop \ge 1} \paren {-1}^k \paren {\binom {n - \paren {k + 1} } k + \binom {n - \paren {k + 1} } {k - 1 } } \paren {2 \cos \theta}^{n - \paren {2 k} } } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sin \theta \paren {\paren {2 \cos \theta}^n + \sum_{k \mathop \ge 1} \paren {-1}^k \binom {n - k } k \paren {2 \cos \theta}^{n - \paren {2 k } } }\) | Pascal's Rule | |||||||||||
\(\ds \) | \(=\) | \(\ds \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - k } k \paren {2 \cos \theta}^{n - \paren {2 k } } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n + 1 - \paren {k + 1} } k \paren {2 \cos \theta}^{n + 1 - \paren {2 k + 1} } }\) | adding and subtracting $1$ |
The result follows by the Principle of Mathematical Induction.
Therefore:
- $\ds \forall n \in \Z_{>0}: \sin n \theta = \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {k + 1} } k \paren {2 \cos \theta}^{n - \paren {2 k + 1} } }$
$\blacksquare$
Examples
Sine of Quintuple Angle
- $\sin 5 \theta = \sin \theta \paren { \paren {2 \cos \theta}^4 - 3 \paren {2 \cos \theta}^2 + 1 }$
Sine of Sextuple Angle
- $\sin 6 \theta = \sin \theta \paren { \paren {2 \cos \theta}^5 - 4 \paren {2 \cos \theta}^3 + 3 \paren {2 \cos \theta} }$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.68$