Tangent of i
Jump to navigation
Jump to search
Theorem
- $\tan i = \paren {\dfrac {e^2 - 1} {e^2 + 1} } i$
where $\tan$ denotes the complex tangent function and $i$ is the imaginary unit.
Proof 1
| \(\ds \tan i\) | \(=\) | \(\ds \frac {\sin i} {\cos i}\) | Definition of Complex Tangent Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\paren {\frac e 2 - \frac 1 {2 e} } i} {\frac e 2 + \frac 1 {2 e} }\) | Sine of $i$ and Cosine of $i$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\frac {e - \frac 1 e} {e + \frac 1 e} } i\) | multiplying denominator and numerator by $2$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\frac {e^2 - 1} {e^2 + 1} } i\) | multiplying denominator and numerator by $e$ |
$\blacksquare$
Proof 2
| \(\ds \tan i\) | \(=\) | \(\ds i \tanh 1\) | Hyperbolic Tangent in terms of Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\frac {e^1 - e^{-1} } {e^1 + e^{-1} } } i\) | Definition of Hyperbolic Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\frac {e^2 - 1} {e^2 + 1} } i\) | multiplying denominator and numerator by $e$ |
$\blacksquare$