Tangent in terms of Hyperbolic Tangent
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Theorem
Let $z \in \C$ be a complex number.
Then:
- $i \tan z = \map \tanh {i z}$
where:
- $\tan$ denotes the tangent function
- $\tanh$ denotes the hyperbolic tangent
- $i$ is the imaginary unit: $i^2 = -1$.
Proof
\(\ds \map \tanh {i z}\) | \(=\) | \(\ds \frac {\map \sinh {i z} } {\map \cosh {i z} }\) | Definition of Hyperbolic Tangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {i \sin z} {\map \cosh {i z} }\) | Sine in terms of Hyperbolic Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {i \sin z} {\cos z}\) | Cosine in terms of Hyperbolic Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds i \tan z\) | Definition of Complex Tangent Function |
$\blacksquare$
Also see
- Sine in terms of Hyperbolic Sine
- Cosine in terms of Hyperbolic Cosine
- Cotangent in terms of Hyperbolic Cotangent
- Secant in terms of Hyperbolic Secant
- Cosecant in terms of Hyperbolic Cosecant
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.82$: Relationship between Hyperbolic and Trigonometric Functions
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $5$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): hyperbolic functions
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): hyperbolic functions