Tag: complex numbers

Questions Related to complex numbers

If $ x+\dfrac{1}{x}=2\cos \theta \   and \ y+\dfrac{1}{y}=2\cos \phi$  then which of the following is not correct?

de moivre’s theorem and its applications demoivre's theorem complex numbers maths

  1. $\displaystyle \frac{x}{y} +\frac{y}{x}=2\cos \left ( \theta -\phi \right )$

  2. $x^{m}y^{n}=\cos \left ( m\theta +n\phi \right )+i\sin \left ( m\theta +n\phi \right )$

  3. $x^{m}y^{n}+x^{-m}y^{-n}=2\cos \left ( m\theta +n\phi \right )$

  4. None of these

Show answer
Correct Option: C
Explanation:

$x+\dfrac { 1 }{ x } =2\cos { \theta  } \quad & \quad y+\dfrac { 1 }{ y } =2\cos { \phi  } \ $

$\Rightarrow x=\cos { \theta  } +i\sin { \theta  } =cis\theta \ \quad & \quad y=\cos { \phi  } +i\sin { \phi  } =cis\phi \ $

$\dfrac { x }{ y } +\dfrac { y }{ x } =\dfrac { cis\theta  }{ cis\phi  } +\dfrac { cis\phi  }{ cis\theta  } =cis\left( \theta -\phi  \right) +cis\left( -\theta +\phi  \right) $

$\therefore \quad \dfrac { x }{ y } +\dfrac { y }{ x } =2\cos { \left( \theta -\phi  \right)  } $

${ x }^{ m }{ y }^{ n }={ \left( cis\theta  \right)  }^{ m }{ \left( cis\phi  \right)  }^{ n }=\left( cism\theta  \right) \left( cisn\phi  \right) $          ...{ De Moivre's Theorem}

$\therefore \quad { x }^{ m }{ y }^{ n }=cis\left( m\theta +n\phi  \right) =\cos { \left( m\theta +n\phi  \right)  } +i\sin { \left( m\theta +n\phi  \right)  } $

${ x }^{ -m }{ y }^{ -n }={ \left( cis\theta  \right)  }^{ -m }{ \left( cis\phi  \right)  }^{ -n }=\left( cis\left( -m\theta  \right)  \right) \left( cis\left( -n\phi  \right)  \right) \ \therefore \quad { x }^{ -m }{ y }^{ -n }=cis\left( -m\theta -n\phi  \right) =\cos { \left( m\theta +n\phi  \right)  } -i\sin { \left( m\theta +n\phi  \right)  } \ $

$\therefore \quad { x }^{ m }{ y }^{ n }+{ x }^{ -m }{ y }^{ -n }=2\cos { \left( m\theta +n\phi  \right)  } $


Let $\mathrm{z}=\cos\theta+\mathrm{i}\sin\theta$. Then the value of $\displaystyle \sum _{\mathrm{m}=1}^{15}{\rm Im}(\mathrm{z}^{2\mathrm{m}-1})$ at $\theta =2^{\mathrm{o}}$ is

de moivre’s theorem and its applications demoivre's theorem complex numbers maths

  1. $\displaystyle \frac{1}{\sin 2^{\mathrm{o}}}$

  2. $\displaystyle \frac{1}{3\sin 2^{\mathrm{o}}}$

  3. $\displaystyle \frac{1}{2\sin 2^{\mathrm{o}}}$

  4. $\displaystyle \frac{1}{4\sin 2^{\mathrm{o}}}$

Show answer
Correct Option: D
Explanation:

$\mathrm{z}=\cos\theta+\mathrm{i}\sin\theta$
$\Rightarrow z^{2m-1} = \cos(2m-1)\theta+i\sin (2m-1)\theta$


Let $X = \displaystyle \sum _{\mathrm{m}=1}^{15}{\rm Im}(\mathrm{z}^{2\mathrm{m}-1})$

$\therefore \mathrm{X}=\sin\theta+\sin 3\theta+\ldots+\sin 29\theta$


$ 2(sin\theta)X=2\sin\theta\sin\theta + 2\sin\theta\sin3\theta...........+2\sin\theta\sin29\theta$
$\Rightarrow  2(\sin\theta)\mathrm{X}=1-\cos 2\theta+\cos 2\theta-\cos 4\theta+\ldots+\cos 28\theta-\cos 30\theta$

$\displaystyle \therefore  \mathrm{X}=\frac{1-\cos 30\theta}{2\sin\theta}=\frac{1}{4\sin 2^{\mathrm{o}}}$