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de Moivre's Formula

Date: 08/11/2002 at 20:38:10
From: Huey Kwik
Subject: de Moivre's formula

How do I use de Moivre's formula to express cos(3p) and sin(3p) in 
terms of cos p and sin p?

And why would this be useful?


Date: 08/12/2002 at 09:23:12
From: Doctor Jaime
Subject: Re: de Moivre's formula

Hello,

If z = a (cos p + i sin p) then z^3 = a^3 (cos 3p + i sin 3p).

So

   a^3 ( cos p + i sin p)^3 = = a^3 ( cos 3p + i sin 3p)

and so

   (cos p + i sin p)^3 = = cos 3p + i sin 3p

If you make the calculations in (cos p + i sin p)^3 you can get your 
conclusion easily.

This is useful if you know the value of cos p and sin p and you want 
to get the values of cos 3p and sin 3p.

It is also useful if you know the values of cos 3p and sin 3p and want
to calculate the values of cos p and sin p. This is more difficult 
because you have to solve a polynomial equation of degree 3, but in 
practice it is more useful.

For example, you know easily the value of cos 30 and sin 30. This way 
you can calculate the not so easy value of cos 10 and sin 10.

Please feel free to ask again if you have more questions.

- Doctor Jaime, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
College Trigonometry
High School Trigonometry

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