Question

If n is any integer, positive negative integer then (cose+ isine )n=cosne+isinne​

Answer 1

De Moivre's theorem

When $n$ is an integer, positive or negative, and $\theta$ is a real number

$\quad (cos\theta+isin\theta)^{n}=cos\:n\theta+i\:sin\:n\theta$;

when $n$ is a fraction, positive or negative, and $\theta$ is a real number

$\quad cos\:n\theta+i\:sin\:n\theta$ is one of the values of $(cos\theta+i\:sin\theta)^{n}$.

Proof:

Case 1. Let $n$ be a positive integer.

• We prove this using Principle of Induction.

✱ When $n=1$,

$\quad (cos\theta+i\:sin\theta)^{1}$

$=cos\theta+i\:sin\theta$

$=cos\:1\theta+i\:sin\:1\theta$

So the given statement is true for $n=1$.

✱ Let the statement be true for $n=m$, where $m$ is a positive integer.

Then $(cos\theta+i\:sin\theta)^{k}=cos\:k\theta+i\:sin\:k\theta$

✱ Now we check whether the statement is true for $n=m+1$.

Therefore $(cos\theta+i\:sin\theta)^{m+1}$

$=(cos\theta+i\:sin\theta)^{m}(cos\theta+i\:sin\theta)^{1}$

$=(cos\:m\theta+i\:sin\:m\theta)(cos\theta+i\:sin\theta)$

$=(cos\:m\theta\:cos\theta-sin\:m\theta\:sin\theta)+i(cos\:m\theta\:sin\theta+sin\:m\theta\:cos\theta)$

$=cos\:(m+1)\theta+i\:sin\:(m+1)\theta$

This shows that the statement holds for $n=m+1$ when the statement is true for $n=m$ and $n=1$.

Thus by the principle of induction, the therorem holds for all the positive integers $n$.

Case 2. Let $n$ be a negative integer.

Let $n=-m$, where $m\in\mathbb{Z}^{+}$

Now $(cos\theta+i\:sin\theta)^{n}$

$=(cos\theta+i\:sin\theta)^{-m}$

$=\dfrac{1}{(cos\theta+i\:sin\theta)^{m}}$

$=\dfrac{1}{cos\:m\theta+i\:sin\:m\theta}$

• We have proved this in Case 1.

$=\dfrac{(cos\:m\theta-i\:sin\:m\theta}{(cos\:m\theta+i\:sin\:m\theta)(cos\:m\theta-i\:sin\:m\theta)}$

$=\dfrac{cos\:m\theta-i\:sin\:m\theta}{cos^{2}m\theta+sin^{2}m\theta}$

$=cos\:m\theta-i\:sin\:m\theta$

$=cos\:(-n)\theta-i\:sin\:(-n)\theta$

• since $n=-m\Rightarrow m=-n$

$=cos\:n\theta+i\:sin\:n\theta$

This proves that

$(cos\theta+i\:sin\theta)^{n}=cos\:n\theta+i\:sin\:n\theta$

is true when $n$ is an integer, positive or negative.