Inverse Hyperbolic Sine of Imaginary Number
Jump to navigation
Jump to search
This article, or a section of it, needs explaining, namely:
Investigate the multifunctional nature of Definition:Complex Inverse Sine and similarly for Hyperbolic Sine
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this point in more detail, feel free to use the talk page.
Theorem
- $\sinh^{-1} \left({i x}\right) = i \sin^{-1} x$
Proof
| \(\displaystyle y\) | \(=\) | \(\displaystyle \sinh^{-1} \left({i x}\right)\) | |||||||||||
| \(\displaystyle \implies \ \ \) | \(\displaystyle \sinh y\) | \(=\) | \(\displaystyle i x\) | Definition of Inverse Hyperbolic Sine | |||||||||
| \(\displaystyle \implies \ \ \) | \(\displaystyle i \sinh y\) | \(=\) | \(\displaystyle - x\) | $i^2 = -1$ | |||||||||
| \(\displaystyle \implies \ \ \) | \(\displaystyle \sin \left({i y}\right)\) | \(=\) | \(\displaystyle - x\) | Hyperbolic Sine in terms of Sine | |||||||||
| \(\displaystyle \implies \ \ \) | \(\displaystyle i y\) | \(=\) | \(\displaystyle \sin^{-1} \left({- x}\right)\) | Definition of Complex Inverse Sine | |||||||||
| \(\displaystyle \implies \ \ \) | \(\displaystyle i y\) | \(=\) | \(\displaystyle -\sin^{-1} x\) | Inverse Sine is Odd Function | |||||||||
| \(\displaystyle \implies \ \ \) | \(\displaystyle y\) | \(=\) | \(\displaystyle i \sinh^{-1} x\) | multiplying both sides by $-i$ |
$\blacksquare$
Investigate the multifunctional nature of Definition:Complex Inverse Sine and similarly for Hyperbolic Sine
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this point in more detail, feel free to use the talk page.
If you are able to explain it, then when you have done so you can remove this instance of {{Explain}} from the code.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.93$: Relationship between Inverse Hyperbolic and Inverse Trigonometric Functions
