Hard induction question (1 Viewer)

[Jackson94]

Prove by induction that (cosθ + isinθ)^n = cos(nθ) + isin(nθ) for all integers n is greater than or equal to 1

 

[httton]

by demoivres theorem

 

[httton]

or what you can do is use angle sum formulas

 

[michaeljennings]

by demoivres theorem

+1

De Moivres Theorem would be the easiest way to do this question

 

[tohriffic]

Let n = 1,
<img src="http://latex.codecogs.com/gif.latex?LHS = (cos\theta + isin\theta)^{1}" title="LHS = (cos\theta + isin\theta)^{1}" />
<img src="http://latex.codecogs.com/gif.latex?= (cos(1\theta) + isin(1\theta))" title="= (cos(1\theta) + isin(1\theta))" />

<img src="http://latex.codecogs.com/gif.latex?RHS = cos (1\theta) + isin(1\theta)" title="RHS = cos (1\theta) + isin(1\theta)" />
<img src="http://latex.codecogs.com/gif.latex?=LHS" title="=LHS" />

Therefore this rule is true for n = 1.

Assume for n = k, the above rule is satisfied.

Required to prove: k+1
<img src="http://latex.codecogs.com/gif.latex?LHS = (cos\theta+isin\theta)^{k+1}" title="LHS = (cos\theta+isin\theta)^{k+1}" />
<img src="http://latex.codecogs.com/gif.latex?= (cos\theta+isin\theta)^{k}(cos\theta+isin\theta)" title="= (cos\theta+isin\theta)^{k}(cos\theta+isin\theta)" />
<img src="http://latex.codecogs.com/gif.latex?=(cosk\theta cos\theta - sink\theta sin\theta) + i (sink\theta cos\theta+ cosk\theta sin\theta)" title="=(cosk\theta cos\theta - sink\theta sin\theta) + i (sink\theta cos\theta+ cosk\theta sin\theta)" />
<img src="http://latex.codecogs.com/gif.latex?=cos(k\theta + \theta) + isin(k\theta + \theta)" title="=cos(k\theta + \theta) + isin(k\theta + \theta)" />
<img src="http://latex.codecogs.com/gif.latex?=cos(k + 1) \theta + isin(k + 1) \theta" title="=cos(k + 1) \theta + isin(k + 1) \theta" />

<img src="http://latex.codecogs.com/gif.latex?RHS = cos (k+1)\theta + isin(k+1)\theta" title="RHS = cos (k+1)\theta + isin(k+1)\theta" />
<img src="http://latex.codecogs.com/gif.latex?=LHS" title="=LHS" />

i.e. it is true for n = k + 1 if it is true for n = k. But since it is true for 1 (k), then it must be true for 2 (k+1) and so on and so on. Therefore, this is generally true for positive integers.

 

[tohriffic]

+1

De Moivres Theorem would be the easiest way to do this question

Lolyeah pretty much.

 

[AAEldar]

http://en.wikipedia.org/wiki/De_Moivre's_formula#Proof_by_induction_.28for_integer_n.29