\(\sinh\) and \(\cosh\)

So I recently found out that

\[\sin {(a + bi)} = i\sinh {(b - ai)}\]

\[\cos {(a + bi)} = \cosh {(b - ai)}\]

and that

\[\sinh {(x)} = \frac {e^x - e^{-x}}{2}\]

\[\cosh {(x)} = \frac {e^x + e^{-x}}{2}\]

Since

\[\frac {a}{b} + \frac {b}{a} = \frac {a^2 + b^2}{ab}\]

\[\frac {a}{b} - \frac {b}{a} = \frac {a^2 - b^2}{ab}\]

and

\[(\frac {a}{b})^{-x} = (\frac {b}{a})^x\]

Re-arranging the equations for \(\sinh{(x)}\) and \(\cosh {(x)}\) gives us

\[\sinh {(x)} = \frac {e^{2x} - 1}{2e^x}\]

\[\cosh {(x)} = \frac {e^{2x} + 1}{2e^x}\]

Substituting these into \(\sin {(a + bi)}\) and \(\cos {(a + bi)}\) when \(x = b - ai\) gives us

\[\sin {(a + bi)} = i\frac {e^{2(b - ai)} - 1}{2e^{b - ai}}\]

\[\cos {(a + bi)} = \frac {e^{2(b - ai)} + 1}{2e^{b - ai}}\]

I'm going to try and expand and substitute by using

\[e^{a + bi} = e^a (\cos {(b)} + i\sin {(b)})\]

Let \(\cos {\theta} + i\sin {\theta} = \) \(cis\) \(\theta\). So

\[e^{a + bi} = e^a \cdot cis (b)\]

Next we substitute this new equation in to get

\[\sin {(a + bi)} = i\frac {e^{2b} \cdot cis (-2a) - 1}{2e^{b} \cdot cis (-a)}\]

\[\cos {(a + bi)} = \frac {e^{2b} \cdot cis (-2a) + 1}{2e^{b} \cdot cis (-a)}\]

So what does \(cis(a + bi)\) equal

Substituting in the two equations gives us

\[cis(a + bi) = \frac {e^{2b} \cdot cis (-2a) + 1}{2e^{b} \cdot cis (-a)} + i(i\frac {e^{2b} \cdot cis (-2a) - 1}{2e^{b} \cdot cis (-a)})\]

That looks unfriendly, let's simplify it

\[cis(a + bi) = \frac {1}{e^{b} \cdot cis(-a)}\]

(Go through the steps, it works)

This can be simplified even further

\[cis(a + bi) = \frac {cis(a)}{e^b}\]

P.s. \[(\cos {\theta} + i\sin {\theta})^n = \cos {\theta n} + i\sin{\theta n}\]

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
`2 \times 3`	\( 2 \times 3 \)
`2^{34}`	\( 2^{34} \)
`a_{i-1}`	\( a_{i-1} \)
`\frac{2}{3}`	\( \frac{2}{3} \)
`\sqrt{2}`	\( \sqrt{2} \)
`\sum_{i=1}^3`	\( \sum_{i=1}^3 \)
`\sin \theta`	\( \sin \theta \)
`\boxed{123}`	\( \boxed{123} \)

Excel in math and science