Product Formula for Sine
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Theorem
- $\ds \map \sin {n z} = 2^{n - 1} \prod_{k \mathop = 0}^{n - 1} \map \sin {z + \frac {k \pi} n}$
Corollary
Let $m \in \Z$ such that $m > 1$.
Then:
- $\ds \prod_{k \mathop = 1}^{m - 1} \sin \frac {k \pi} m = \frac m {2^{m - 1} }$
Proof
We have:
| \(\ds \map \sin {n z}\) | \(=\) | \(\ds \frac {e^{i n z} - e^{-i n z} } {2 i}\) | Euler's Sine Identity | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^{i n z} - e^{-i n z} } {2 i} \times \frac {e^{i n z} } {e^{i n z} } \times \frac {2^n} {2^n}\) | multiplying by $1$ | |||||||||||
| \(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds 2^{n - 1} \frac {e^{2 i n z} - 1} {2^n i e^{i n z} }\) |
Consider the equation:
- $x^n - 1 = 0$
whose solutions are the complex roots of unity:
- $1, e^{-2 \pi i / n}, e^{-4 \pi i / n}, e^{-6 \pi i / n}, \ldots, e^{-2 \paren {n - 1} \pi i / n}$
Then:
| \(\ds x^n - 1\) | \(=\) | \(\ds \paren {x - 1} \paren {x - e^{-2 \pi i / n} } \paren {x - e^{-4 \pi i / n} } \dotsm \paren {x - e^{-2 \paren {n - 1} \pi i / n} }\) | product of all the roots |
Let $x = e^{2 i z}$.
Then:
| \(\ds e^{2 i nz} - 1\) | \(=\) | \(\ds \paren {e^{2 i z} - 1} \paren {e^{2 i z} - e^{-2 \pi i / n} } \paren {e^{2 i z} - e^{-4 \pi i / n} } \dotsm \paren {e^{2 i z} - e^{-2 \paren {n - 1} \pi i / n} }\) | product of all the roots | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {e^{2 i z} - 1} \paren {e^{2 i z} - e^{-2 \pi i / n} } \paren {\dfrac {e^{2 \pi i / n} } {e^{2 \pi i / n} } } \paren {e^{2 i z} - e^{-4 \pi i / n} } \paren {\dfrac {e^{4 \pi i / n} } {e^{4 \pi i / n} } } \dotsm \paren {e^{2 i z} - e^{-2 \paren {n - 1} \pi i / n} } \paren {\dfrac {e^{2 \paren {n - 1} \pi i / n} } {e^{2 \paren {n - 1} \pi i / n} } }\) | multiplying by $1$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\paren {e^{2 i z} - 1} \paren {e^{2 i \paren {z + \dfrac \pi n} } - 1} \paren {e^{2 i \paren {z + \dfrac {2 \pi} n} } - 1} \dotsm \paren {e^{2 i \paren {z + \dfrac {\paren {n - 1} \pi} n} } - 1} } {e^{2 \pi i / n} e^{4 \pi i / n} \dotsm e^{2 \paren {n - 1} \pi i / n} }\) | Product of Powers | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds \) | \(=\) | \(\ds \dfrac {\paren {e^{2 i z} - 1} \paren {e^{2 i \paren {z + \dfrac \pi n} } - 1} \paren {e^{2 i \paren {z + \dfrac {2 \pi} n} } - 1} \dotsm \paren {e^{2 i \paren {z + \dfrac {\paren {n - 1} \pi} n} } - 1} } {\paren {-1}^{\paren {n - 1} } }\) | Product of nth Roots of Unity |
Plugging $\paren {2}$ into $\paren {1}$, we have:
| \(\ds \map \sin {n z}\) | \(=\) | \(\ds 2^{n - 1} \frac {e^{2 i n z} - 1} {2^n i e^{i n z} }\) | From $\paren {1}$ above | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2^{n - 1} \dfrac {\paren {e^{2 i z} - 1} \paren {e^{2 i \paren {z + \dfrac \pi n} } - 1} \paren {e^{2 i \paren {z + \dfrac {2 \pi} n} } - 1} \dotsm \paren {e^{2 i \paren {z + \dfrac {\paren {n - 1} \pi} n} } - 1} } {2^n i e^{i n z} \paren {-1}^{\paren {n - 1} } }\) | Plugging $\paren {2}$ into $\paren {1}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2^{n - 1} \dfrac {\paren {e^{2 i z} - 1} \paren {e^{2 i \paren {z + \dfrac \pi n} } - 1} \paren {e^{2 i \paren {z + \dfrac {2 \pi} n} } - 1} \dotsm \paren {e^{2 i \paren {z + \dfrac {\paren {n - 1} \pi} n} } - 1} } {2^n i e^{i n z} \paren {i^2}^{\paren {n - 1} } }\) | Powers of Imaginary Unit | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2^{n - 1} \dfrac {\paren {e^{2 i z} - 1} \paren {e^{2 i \paren {z + \dfrac \pi n} } - 1} \paren {e^{2 i \paren {z + \dfrac {2 \pi} n} } - 1} \dotsm \paren {e^{2 i \paren {z + \dfrac {\paren {n - 1} \pi} n} } - 1} } {2^n i^{2 n - 1} e^{i n z} }\) | Product of Powers | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2^{n - 1} \dfrac {\paren {e^{2 i z} - 1} \paren {e^{2 i \paren {z + \dfrac \pi n} } - 1} \paren {e^{2 i \paren {z + \dfrac {2 \pi} n} } - 1} \dotsm \paren {e^{2 i \paren {z + \dfrac {\paren {n - 1} \pi} n} } - 1} } {2^n i^n e^{i n z} \paren {e^{i \pi / 2} }^{n - 1} }\) | $e^{i \pi / 2} = i$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2^{n - 1} \dfrac {\paren {e^{2 i z} - 1} \paren {e^{2 i \paren {z + \dfrac \pi n} } - 1} \paren {e^{2 i \paren {z + \dfrac {2 \pi} n} } - 1} \dotsm \paren {e^{2 i \paren {z + \dfrac {\paren {n - 1} \pi} n} } - 1} } {2^n i^n e^{i \paren {nz + \dfrac {\pi \paren {\paren {n - 1} n} } {2 n} } } }\) | Product of Powers | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2^{n - 1} \dfrac {\paren {e^{2 i z} - 1} \paren {e^{2 i \paren {z + \dfrac \pi n} } - 1} \paren {e^{2 i \paren {z + \dfrac {2 \pi} n} } - 1} \dotsm \paren {e^{2 i \paren {z + \dfrac {\paren {n - 1} \pi} n} } - 1} } {2^n i^n e^{i \paren {nz + \dfrac {\pi \paren {1 + 2 + \dotsm + \paren {n - 1} } } n} } }\) | Closed Form for Triangular Numbers | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2^{n - 1} \paren {\dfrac {e^{2 i z} - 1} {2 i e^{i z} } } \paren {\dfrac {e^{2 i \paren {z + \dfrac \pi n} } - 1} {2 i e^{i \paren {z + \dfrac \pi n} } } } \paren {\dfrac {e^{2 i \paren {z + \dfrac {2 \pi} n} } - 1} {2 i e^{i \paren {z + \dfrac {2 \pi} n} } } } \dotsm \paren {\dfrac {e^{2 i \paren {z + \dfrac {\paren {n - 1} \pi} n} } - 1} {2 i e^{i \paren {z + \dfrac {\paren {n - 1} \pi} n} } } }\) | Product of Powers | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2^{n - 1} \paren {\dfrac {e^{i z} - e^{-i z} } {2 i} } \paren {\dfrac {e^{i \paren {z + \dfrac \pi n} } - e^{-i \paren {z + \dfrac \pi n} } } {2 i} } \paren {\dfrac {e^{i \paren {z + \dfrac {2 \pi} n} } - e^{-i \paren {z + \dfrac {2 \pi} n} } } {2 i} } \dotsm \paren {\dfrac {e^{i \paren {z + \dfrac {\paren {n - 1} \pi} n} } - e^{-i \paren {z + \dfrac {\paren {n - 1} \pi} n} } } {2 i} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2^{n - 1} \prod_{k \mathop = 0}^{n - 1} \map \sin {z + \dfrac {k \pi} n}\) | Euler's Sine Identity |
$\blacksquare$