Frank's questions at Yahoo Answers regarding de Moivre's theorem

[MarkFL]

Here are the questions:

Induction and Complex #'s. Calc help?


a) Prove, using mathematical induction, that for a positive integer n, (cos(x) + isinx)^n = cosnx +i sinnx where I^2 + -1

b)The complex number z is defined by z = cosx + isinx
I) show that 1/z = cos (-x) + isin(-x)
II) Deduce that z^n + z^-n = 2cosnx

c) Find the binomial expansion of (z + z^-1)^5
I) Hence show tat cos^5x = 1/16(acoas5x + bcos3x + ccosx) where a,b,c are positive integers to be found.

Thank you so much for you help!

I have posted a link there to this topic, so the OP can see my work.

 

[MarkFL]

Hello Frank,

a) First, we check to see if $P_1$ (the case where $n=1$) is valid.

$$\left( \cos( \theta)+i \cdot \sin( \theta) \right)^1= \cos(1 \cdot \theta)+i \cdot \sin(1 \cdot \theta)$$

$$\cos( \theta)+i \cdot \sin( \theta)= \cos( \theta)+i \cdot \sin( \theta)$$

$P_1$ is true. Next, our induction hypothesis $P_n$ is:

$$\left( \cos( \theta)+i \cdot \sin( \theta) \right)^n= \cos(n \cdot \theta)+i \cdot \sin(n \cdot \theta)$$

Multiply both sides by $$\cos( \theta)+i \cdot \sin( \theta)$$:

$$\left( \cos( \theta)+i \cdot \sin( \theta) \right)^n \left( \cos( \theta)+i \cdot \sin( \theta) \right)= \left( \cos(n \cdot \theta)+i \cdot \sin(n \cdot \theta) \right) \left( \cos( \theta)+i \cdot \sin( \theta) \right)$$

$$ \left( \cos( \theta)+i \cdot \sin( \theta) \right)^{n+1}= \cos(n \cdot \theta) \cos( \theta)+i \cdot \cos(n \cdot \theta) \sin( \theta)+i \cdot \cos( \theta) \sin(n \cdot \theta)+i^2 \cdot \sin( \theta) \sin(n \cdot \theta)$$

$$\left( \cos( \theta)+i \cdot \sin( \theta) \right)^{n+1}= \left( \cos(n \cdot \theta) \cos( \theta)- \sin(n \cdot \theta) \sin( \theta) \right)+i \left( \sin( \theta) \cos(n \cdot \theta)+ \cos( \theta) \sin(n \cdot \theta) \right)$$

$$\left( \cos( \theta)+i \cdot \sin( \theta) \right)^{n+1}= \cos \left(n \cdot \theta+ \theta \right)+i \cdot \sin \left(n \cdot \theta+ \theta \right)$$

$$\left( \cos(\theta)+i \cdot \sin( \theta) \right)^{n+1}= \cos \left((n+1) \cdot \theta \right)+i \cdot \sin \left((n+1) \cdot \theta \right)$$

We have derived $P_{n+1}$ from $P_n$ thereby completing the proof by induction.

b) For this question and for part c), we will find proving the following useful:

$$\left( \cos( \theta)+i \cdot \sin( \theta) \right)^{-n}= \cos(n \cdot \theta)-i \cdot \sin(n \cdot \theta)$$ where $n\le0\in\mathbb{Z}$

First, we check to see if $P_0$ (the case where $n=0$) is valid.

$$\left( \cos( \theta)+i \cdot \sin( \theta) \right)^0= \cos(0 \cdot \theta)+i \cdot \sin(0 \cdot \theta)$$

$$1= \cos(0)+i \cdot \sin(0)=1$$

$P_0$ is true. Next, our induction hypothesis $P_n$ is:

$$\left( \cos( \theta)+i \cdot \sin( \theta) \right)^{-n}= \cos(n \cdot \theta)-i \cdot \sin(n \cdot \theta)$$

Multiply both sides by $$\left(\cos( \theta)+i \cdot \sin( \theta) \right)^{-1}$$:

$$\left( \cos( \theta)+i \cdot \sin( \theta) \right)^{-n} \left( \cos( \theta)+i \cdot \sin( \theta) \right)^{-1}= \frac{\cos(n \cdot \theta)-i \cdot \sin(n \cdot \theta)}{ \cos( \theta)+i \cdot \sin( \theta)}$$

$$\left( \cos( \theta)+i \cdot \sin( \theta) \right)^{-(n+1)}= \frac{ \cos(n \cdot \theta)-i \cdot \sin(n \cdot \theta)}{ \cos( \theta)+i \cdot \sin( \theta)} \cdot \frac{ \cos( \theta)-i \cdot \sin( \theta)}{ \cos( \theta)-i \cdot \sin( \theta)}$$

$$\left( \cos( \theta)+i \cdot \sin( \theta) \right)^{-(n+1)}= \frac{ \cos(n \cdot \theta) \cos(\theta)-i \cos(n \cdot \theta) \sin(\theta)-i \sin(n \cdot \theta) \cos(\theta)+i^2 \sin(n \cdot \theta) \sin(\theta)}{\cos^2(\theta)+\sin^2(\theta)}$$

$$\left( \cos( \theta)+i \cdot \sin( \theta) \right)^{-(n+1)}= \left(\cos(n \cdot \theta) \cos(\theta)- \sin(n \cdot \theta) \sin(\theta) \right)-i \left(\cos(n \cdot \theta) \sin(\theta)+ \sin(n \cdot \theta) \cos(\theta) \right)$$

$$\left( \cos( \theta)+i \cdot \sin( \theta) \right)^{-(n+1)}= \cos \left((n+1)\theta \right)-i \cdot \sin \left((n+1)\theta \right)$$

We have derived $P_{n+1}$ from $P_n$ thereby completing the proof by induction.

Observing that $\cos(-\theta)=\cos(\theta)$ and $\sin(-\theta)=-\sin(\theta)$, we may state:

$$\left( \cos( \theta)+i \cdot \sin( \theta) \right)^{-n}= \cos(-n \cdot \theta)+i \cdot \sin(-n \cdot \theta)$$

Thus, we have proved:

$$\left( \cos( \theta)+i \cdot \sin( \theta) \right)^{k}= \cos(k \cdot \theta)+i \cdot \sin(k \cdot \theta)$$ where $$k\in\mathbb{Z}$$

i) Given that $$z=\cos(x)+i \cdot \sin(x)$$ then we may use our theorem (actually de Moivre's theorem) to state:

$$\frac{1}{z}=z^{-1}=\left(\cos(x)+i \cdot \sin(x) \right)^{-1}=\cos(-x)+i \cdot \sin(-x)$$

ii) And so we may show that:

$$z^n+z^{-n}=\cos(nx)+i \cdot \sin(nx)+\cos(x)-i \cdot \sin(nx)=2 \cos(nx)$$

c) Using the binomial theorem, we find:

$$\left(z+z^{-1} \right)^5=\sum_{k=0}^5{5 \choose k}z^{5-k}z^{-k}=\sum_{k=0}^5{5 \choose k}z^{5-2k}$$

$$\left(z+z^{-1} \right)^5=z^5+5z^3+10z+10z^{-1}+5z^{-3}+z^{-5}$$

$$\left(z+z^{-1} \right)^5=\left(z^5+z^{-5} \right)+5\left(z^3+z^{-3} \right)+10\left(z+z^{-1} \right)$$

Using the result of b) ii) we may state:

$$\left(z+z^{-1} \right)^5=\left(2 \cos(5x) \right)+5\left(2 \cos(3x) \right)+10\left(2 \cos(x) \right)$$

$$\left(z+z^{-1} \right)^5=2 \cos(5x)+10 \cos(3x)+20 \cos(x)$$

i) Using the result of b) ii) we may state:

$$\left(z+z^{-1} \right)^5=\left(2\cos(x) \right)^5=32\cos^5(x)$$

And so using the previous result, we then find:

$$32\cos^5(x)=2 \cos(5x)+10 \cos(3x)+20 \cos(x)$$

Hence:

$$\cos^5(x)=\frac{1}{16}\left( \cos(5x)+5 \cos(3x)+10 \cos(x) \right)$$

Thus, we have found:

$$a=1,\,b=5,\,c=10$$