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Subject: "Trigonometric identities with complex numbers"     Previous Topic | Next Topic
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Gerenuk
Member since Oct-13-09
 Oct-13-09, 03:35 PM (EST)
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"Trigonometric identities with complex numbers"
 
   It is well known that powers of trigonometric functions can easily be derived with cos(x)=Re(exp(ix)) and de Moivre. But I'm not sure if everyone is aware that almost all identities can be derived this way in a *systematic* way! Only fractions cause some trouble.

I attach one page of my Math notes where I wrote that down.

If anyone has suggestions how to deal to tan(x) or fractions in identities, then comments are welcome.

What do you think?

Attachments
https://www.cut-the-knot.org/htdocs/dcforum/User_files/4ad4e10f5392c1f6.jpg

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alexbadmin
Charter Member
2452 posts		 Oct-14-09, 06:54 AM (EST)
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1. "RE: Trigonometric identities with complex numbers"
In response to message #0
 
   I am pretty sure that there are no nice or generic identities for trigonometric fractions. At least, I am unaware of any. Never saw a formula for tan(5x), say.

But what you did with integer powers looks looks just beautiful. I have a vague memory of seeing something similar but can't at the moment put a finger on the source.


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Gerenuk
Member since Oct-13-09
 Oct-14-09, 11:29 AM (EST)
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2. "RE: Trigonometric identities with complex numbers"
In response to message #1
 
   The best thing is, that it's like differentiation: (if there are no fractions then) the bringing in/out rules give you enough power to arrive at any form without being stuck.

Yet, there are also identities with fractions that can be dealt with the same system, but not in a systematic way I believe.

In the attachment I show some identities.

Hmm... file uploading in this forum seems to be broken today... so here is a like to the picture:
https://yfrog.com/1qmathsgj


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alexbadmin
Charter Member
2452 posts		 Oct-16-09, 09:43 PM (EST)
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3. "RE: Trigonometric identities with complex numbers"
In response to message #2
 
   > Hmm... It may be the size. I keep it under 50K.

There is nothing remotely as wonderful for fractions is fo rpositive powers. The formalism is actually fun when you get the hang of it.


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