# De Moivre's Theorem

• Aug 19th 2012, 07:53 PM
ok59
[Solved] De Moivre's Theorem

if a=cos2alpha + isin2alpha
b=cos2beta + isin2beta
prove that,

a-b / a+b = itan (alpha-beta)
• Aug 19th 2012, 08:15 PM
richard1234
Re: De Moivre's Theorem
Assuming "2alpha" is $2 \alpha$, then

$a = e^{2i \alpha}$
$b = e^{2i \beta}$

Then, $\frac{a-b}{a+b} = \frac{e^{2i \alpha} - e^{2i \beta}}{e^{2i \alpha} + e^{2i \beta}}$. Can you take it from there?
• Aug 19th 2012, 08:23 PM
ok59
Re: De Moivre's Theorem
sorry i dont get it ( still learning )
how do i suppose to get tan from it.
do you mean (cos alpha + isin alpha)^2
• Aug 19th 2012, 08:59 PM
richard1234
Re: De Moivre's Theorem
Quote:

Originally Posted by ok59
sorry i dont get it ( still learning )
how do i suppose to get tan from it.
do you mean (cos alpha + isin alpha)^2

Yes, it's also equal to $(\cos \alpha + i \sin \alpha)^2$ by de Moivre's. I don't know if that helps though. Expanding doesn't yield anything useful.

It could also help to play around with the RHS, $i \tan (\alpha - \beta)$. Note that this is a purely imaginary number. You can try to reduce the LHS as $i \frac{\sin (\alpha - \beta)}{\cos (\alpha - \beta)}$ or something like that. You'll most probably need your trigonometric identities.
• Aug 19th 2012, 09:08 PM
ok59
Re: De Moivre's Theorem
hmm.. this problem is so freaking weird . I am about to give up ( it has been 2 days).
• Aug 19th 2012, 09:36 PM
richard1234
Re: De Moivre's Theorem
Yeah, it's an interesting problem. I haven't solved it yet...there seem to be several possible ways to the solution though. Just try a bunch of methods, see if one of them works. Maybe another user has a nice, elegant solution that doesn't require a lot of brute force (I don't want to brute force algebra/trig identities right now...). There might even be a geometric solution involving two points on a circle and finding the tangent of the angle in between.
• Aug 19th 2012, 10:22 PM
Deveno
Re: De Moivre's Theorem
let's write, for the time being, cos(2α) = u, sin(2α) = v, cos(2β) = y, sin(2β) = z.

then:

$\frac{a-b}{a+b} = \frac{u-y + i(v-z)}{u+y + i(v+z)} = \frac{((u-y) + i(v-z))((u+y) - i(v+z))}{(u+y)^2 + (v+z)^2}$

$= \frac{(u-y)(u+y) + (v-z)(v+z) + i(-(u-y)(v+z) + (v-z)(u+y))}{u^2 + 2uy + y^2 + v^2 + 2vz + z^2}$

$=\frac{u^2 + v^2 - y^2 - z^2 + i(2(vy - uz))}{u^2 + v^2 + y^2 + z^2 + 2(uy + vz)}$

now we can use some trig to simplify.

note that u2+v2 = y2+z2 = 1. so our messy fraction becomes:

$\frac{1 - 1 + i(2(vy - uz))}{2 + 2(uy + vz))} = i\frac{vy - uz}{1 + uy + vz}$

now $vy - uz = \sin(2\alpha)\cos(2\beta) - \cos(2\alpha)\sin(2\beta) = \sin(2\alpha - 2\beta)$

and $uy + vz = \cos(2\alpha)\cos(2\beta) + \sin(2\alpha)\sin(2\beta) = \cos(2\alpha - 2\beta)$.

so what we have is:

$i\frac{\sin(2\alpha - 2\beta)}{1 + \cos(2\alpha - 2\beta)} = i\tan\left(\frac{2\alpha - 2\beta}{2}\right) = i\tan(\alpha - \beta)$
• Aug 20th 2012, 06:36 AM
ok59
Re: De Moivre's Theorem
Man ! Thank so very much Deveno.
You made it so simple to understand mate. You can't imagine the time i've spent on this problem.
Thank You again for your help.
Have a nice day !

edit: Thank you to Richard1234 too. I have solved it by your way too. Now I have two solutions.(Cool)
• Aug 20th 2012, 06:45 AM
Soroban
Re: De Moivre's Theorem (Part 1)
Hello, ok59!

Out of curiosity, I brute-forced it . . . whew!

We need the Sum-to-Product Identities:

. . $\sin\!A\;+\;\sin\!B \:=\: 2\sin\!\left(\tfrac{A+B}{2}\right)\cos\!\left( \tfrac{A-B}{2}\right) \qquad \cos\!A\;+\;\cos\!B \:=\: 2\cos\!\left(\tfrac{A+B}{2}\right)\cos\!\left( \tfrac{A-B}{2}\right)$

. . $\sin\!A\;-\;\sin\!B \:=\:2\cos\!\left(\tfrac{A+B}{2}\right)\sin\!\left ( \tfrac{A-B}{2}\right) \qquad \cos\!A\;-\;\cos\!B \:=\:\text{-}2\sin\!\left(\tfrac{A+B}{2}\right)\sin\!\left( \tfrac{A-B}{2}\right)$

Quote:

$\text{Given: }\:\begin{Bmatrix}a &=& \cos2A + i\sin2A \\ b &=& \cos2B + i\sin2B \end{Bmatrix}$

$\text{Prove that: }\:\frac{a-b}{a+b} \:=\; i\tan(A-B)$

$\text{The numerator is: }\:a - b \:=\:(\cos2A-\cos2B) + i(\sin2A - \sin 2B)$

x x . . . . . . . . . . . . . . . $=\;\text{-}2\sin(A\!+\!B)\sin(A\!-\!B) + 2i\cos(A\!+\!B)\sin(A\!-\!B)$

x x . . . . . . . . . . . . . . . $=\;2\sin(A\!-\!B)\bigg[\text{-}\sin(A\!+\!B) + i\cos(A\!+\!B)\bigg]$

$\text{The denominator is: }\:a + b \:=\:(\cos2A + \cos2B) + i(\sin2A +\sin2B)$

. . . . . . . . . . . . . . . . . . . $=\;2\cos(A\!+\!B)\cos(A\!-\!B) + 2i\sin(A\!+\!B)\cos(A\!-\!B)$

. . . . . . . . . . . . . . . . . . . $=\;2\cos(A\!-\!B)\bigg[\cos(A\!+\!B) + i\sin(A\!+\!B)\bigg]$

$\text{And we have: }\:\frac{a-b}{a+b} \;=\;\frac{\sin(A\!-\!B)\bigg[\text{-}\sin(A\!+\!B) + i\cos(A\!+\!B)\bigg]}{\cos(A\!-\!B)\bigg[\cos(A\!+\!B) + i\sin(A\!+\!B)\bigg]}$

. . . . . . . . . . . . . . . $=\;\tan(A\!-\!B)\left[\frac{\text{-}\sin(A\!+\!B) + i\cos(A\!+\!B)}{\cos(A\!+\!B) + i\sin(A\!+\!B)\right]}$

$\text{Rationalize: }\:\tan(A-B)\cdot\left[\frac{\text{-}\sin(A\!+\!B) + i\cos(A\!+\!B)}{\cos(A\!+\!B) + i\sin(A\!+\!B)}\right]\cdot\left[\frac{\cos(A\!+\!B) - i\sin(A\!+\!B)}{\cos(A\!+\!B) - i\sin(A\!+\!B)}\right]$

. . . . . . . $=\;\tan(A-B)\cdot \left[\frac{\text{-}\sin(A\!+\!B)\cos(A\!+\!B) + i\sin^2(A\!+\!B) + i\cos^2(A\!+\!B) - i^2\sin(A\!+\!B)\cos(A\!+\!B)} {\cos^2(A\!+\!B) - i^2\sin^2(A\!+\!B)}\right]$

. . . . . . . $=\;\tan(A-B)\cdot\left[\frac{\text{-}\sin(A\!+\!B)\cos(A\!+\!B) + i\sin^2(A\!+\!B) + i\cos^2(A\!+\!B) + \sin(A\!+\!B)\cos(A\!+\!B)}{\cos^2(A\!+\!B) + \sin^2(A\!+\!B)}\right]$

. . . . . . . $=\;\tan(A-B)\cdot\left[\frac{i\sin^2(A+B) + i\cos^2(A+B)}{1}\right]$

. . . . . . . $=\;\tan(A-B)\cdot i\left[\sin^2(A+B) + \cos^2(A+B)\right]$

. . . . . . . $=\;\tan(A-B)\cdot i \cdot 1$

. . . . . . . $=\;i\tan(A-B)$
• Aug 20th 2012, 12:24 PM
ok59
Re: De Moivre's Theorem (Part 1)
Thanks Mr. Soroban for another method . Now I understand that the trick is to divide and multiply with the conjugate of denominator to get the result. I appreciate the efforts of the members of this board.
• Sep 12th 2012, 10:39 PM
manoj9585
Re: De Moivre's Theorem
De Moivre's Theorem is a relatively simple formula for calculating powers of complex numbers.De Moivre's formula states that for any real number x and any integer n, (cosx + isinx)n = cos(nx) + isin(nx).(r cisθ)^n = r^n cis(nθ) here n is an integer.
free algebra word problems
• Sep 13th 2012, 02:04 AM
johnsomeone
Re: [Solved] De Moivre's Theorem
Here's another (equivalent) way: I'll begin with a general calculation that I'll then apply to this specific problem.

Suppose $z \in \mathbb{C} \ni z \neq -1$, and $|z|^2 = 1$ (i.e. $z$ on the unit circle except the leftmost point).

So let $z = e^{\theta i}$, where $\theta$ is not an odd multiple of $\pi$.

Then $\frac{z-1}{z+1}$

$= \frac{(z-1)(\bar{z}+1)}{(z+1)(\bar{z}+1)} = \frac{|z|^2+z-\bar{z}-1}{|z|^2+z+\bar{z}+1} = \frac{1+z-\bar{z}-1}{1+z+\bar{z}+1}$

$= \frac{z-\bar{z}}{2+(z+\bar{z})} = \frac{2Im(z)i}{2+2Re(z)} = \frac{Im(z)}{1+Re(z)}i = \frac{\sin(\theta)}{1+\cos(\theta)}i = \tan(\theta / 2)i$.

With that in hand, the result follows quickly. Let $a=e^{2\alpha i}, b=e^{2\beta i}$, $a+b \neq 0$. Let $z = a/b$. (Note that $|b| = 1$, so $b \neq 0$.)

Let $\theta = 2(\alpha - \beta)$. Then $z = e^{\theta i}$, and obviously $|z| = 1$.

Notice that from $a+b \neq 0$, it follows that $a \neq -b$, so that $z = a/b \neq -1$, and hence so can apply the above derivation with this $z$.

Now consider $\frac{a-b}{a+b}$. Have:

$\frac{a-b}{a+b} = \frac{a/b - 1}{a/b + 1} = \frac{z-1}{z+1} = \tan(\theta / 2)i = \tan((2(\alpha - \beta)) / 2)i = \tan(\alpha - \beta)i$ as desired.

Thus the claim is established.

Notice that all the "bad" cases amounted to $a + b = 0$, because then $a = -b = e^{(2n+1) \pi i}b$, so $e^{2\alpha i} = e^{(2n+1) \pi i}e^{2\beta i}$, so $\theta = 2(\alpha - \beta) = (2n+1) \pi$.

That would've made $z = e^{\theta i} = e^{(2n+1) \pi i} = -1$, and also would've maded $\alpha - \beta = (n+1/2) \pi$, which is where $\tan(\alpha - \beta)i$ is undefined.