Euler's Sine Identity/Real Domain
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Theorem
For any real number $x \in \R$:
- $\sin x = \dfrac {e^{i x} - e^{-i x} } {2 i}$
where:
- $e^{i x}$ denotes the exponential function
- $\sin x$ denotes the real sine function
- $i$ denotes the inaginary unit.
Proof 1
Recall the definition of the sine function:
\(\ds \sin x\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x - \frac {x^3} {3!} + \frac {x^5} {5!} - \frac {x^7} {7!} + \cdots + \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!} + \cdots\) |
Recall the definition of the exponential as a power series:
\(\ds e^x\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 + \frac x {1!} + \frac {x^2} {2!} + \frac {x^3} {3!} + \cdots + \frac {x^n} {n!} + \cdots\) |
Then, starting from the right hand side:
\(\ds \frac {e^{i x} - e^{-i x} } {2 i}\) | \(=\) | \(\ds \frac 1 {2 i} \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {i x}^n} {n!} - \sum_{n \mathop = 0}^\infty \frac {\paren {-i x}^n} {n!} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 i} \sum_{n \mathop = 0}^\infty \paren {\frac {\paren {i x}^n - \paren {-i x}^n} {n!} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 i} \sum_{n \mathop = 0}^\infty \paren {\frac {\paren {i x}^{2 n} - \paren {-i x}^{2 n} } {\paren {2 n}!} + \frac {\paren {i x}^{2 n + 1} - \paren {-i x}^{2 n + 1} } {\paren {2 n + 1}!} }\) | split into even and odd $n$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 i} \sum_{n \mathop = 0}^\infty \frac {\paren {i x}^{2 n + 1} - \paren {-i x}^{2 n + 1} } {\paren {2 n + 1}!}\) | as $\paren {-i x}^{2 n} = \paren {i x}^{2 n}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 i} \sum_{n \mathop = 0}^\infty \frac {2 \paren {i x}^{2 n + 1} } {\paren {2 n + 1}!}\) | as $\paren {-1}^{2 n + 1} = -1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 i \sum_{n \mathop = 0}^\infty \frac {\paren {i x}^{2 n + 1} } {\paren {2 n + 1}!}\) | cancel $2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 i \sum_{n \mathop = 0}^\infty \frac {i \paren {-1}^n x^{2 n + 1} } {\paren {2 n + 1}!}\) | as $i^{2 n + 1} = i \paren {-1})^n $ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1!} }\) | cancel $i$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sin x\) |
$\blacksquare$
Proof 2
Recall Euler's Formula:
- $e^{i x} = \cos x + i \sin x$
Then, starting from the right hand side:
\(\ds \frac {e^{i x} - e^{-i x} }{2 i}\) | \(=\) | \(\ds \frac {\paren {\cos x + i \sin x} - \paren {\map \cos {-x} + i \map \sin {-x} } } {2 i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {\cos x + i \sin x - \cos x - i \map \sin {-x} } } {2 i}\) | Cosine Function is Even | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {i \sin x - i \map \sin {-x} } {2 i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {i \sin x - i \paren {-\map \sin {-x} } } {2 i}\) | Sine Function is Odd | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 i \sin x} {2 i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sin x\) |
$\blacksquare$
Proof 3
\(\text {(1)}: \quad\) | \(\ds e^{i x}\) | \(=\) | \(\ds \cos x + i \sin x\) | Euler's Formula | ||||||||||
\(\text {(2)}: \quad\) | \(\ds e^{-i x}\) | \(=\) | \(\ds \cos x - i \sin x\) | Euler's Formula: Corollary | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds e^{i x} - e^{-i x}\) | \(=\) | \(\ds \paren {\cos x + i \sin x} - \paren {\cos x - i \sin x}\) | $(1) - (2)$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 2 i \sin x\) | simplifying | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {e^{i x} - e^{-i x} } {2 i}\) | \(=\) | \(\ds \sin x\) |
$\blacksquare$
Also presented as
Euler's Sine Identity can also be presented as:
- $\sin z = \dfrac 1 2 i \paren {e^{-i z} - e^{i z} }$
Also see
- Euler's Cosine Identity
- Euler's Tangent Identity
- Euler's Cotangent Identity
- Euler's Secant Identity
- Euler's Cosecant Identity
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Relationship between Exponential and Trigonometric Functions: $7.17$