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Algebra Level 3

Find the remainder of the division of ${ (\cos { a } +x\sin { a } ) }^{ n }$ by ${ x }^{ 2 }+1$ where $a$ is real and $n$ is a positive integer.

\cos { na } +x\sin { na }

\cos ^{ n }{ a } +x\sin ^{ n }{ a }

x\sin { na }

\cos { 2na } +x\sin { 2na }

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Md Zuhair
Apr 12, 2017

Relevant wiki: De Moivre's Theorem - Raising to a Power - Basic

Quite simple,

By remainder theorem , $f(x)=x^2+1=0$ $\implies$ $x = \sqrt{-1}$ $\implies x = i$ when inserted in the above expression , we get,

$(\cos a + i \sin a)^n$ $\implies$ $e^{ian}$ $\implies$ $\cos an + i \sin an$ $\space$ [ $\text{See Note}$ ]

Note

In complex algebra if a complex number is defined as $z = \cos \theta + i \sin \theta$ , then in euler form it can be written as $z = e^{i \theta}$ . This is the fact used in the above problem to solve it. The answer to the above problem can also be given by the law, known as $\text{De Moivre's Law}$

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