DeMoivre's Theorem 1

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Michael Proudman

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Michael Proudman 7 years, 8 months ago

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Question statement

Use De Moivre's theorem to calculate $(\simplify{+{x}+{y}i})^\var{n}$

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			Name	Type	Generated Value

x	integer	10
y	integer	-1
n	integer	10
theta	decimal	dec("-9.966865249116202737844611987802059024328e-2")
mod	number	10.0498756211
x2	number	5707904498.999998
y2	number	-8825080099.999996

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2. Gap Gap 1. Number entry
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$(\simplify{{x}+{y}i})^\var{n}=A+Bi$

Calculate $A$ and $B$

$A$ = Gap 0.

$B$ = Gap 1.

Advice

To evaluate $(\simplify{{x}+{y}i})^\var{n}$ we must first express $(\simplify{{x}+{y}i})$ in polar form.

The modulus of $\simplify{+{x}+{y}i}$ = $\sqrt{\var{x}^2+(\var{y})^2}=\sqrt{\simplify{{x^2}+{y}^2}}$

The argument of the complex number is given by $\theta=\tan^{-1}\left(\frac{\var{y}}{\var{x}}\right)=\var{theta}$

According to De Moivre's theorem $Z^{\var{n}}=|Z|^{\var{n}}\left(\cos(\var{n}*\theta)+i\sin(\var{n}*\theta)\right)$

$(\simplify{{x}+{y}i})^\var{n}=\left(\sqrt{\simplify{{x^2}+{y}^2}}\right)^{\var{n}}\left(\cos(\simplify{{n}*{theta}})+i\sin(\simplify{{n}*{theta}})\right)$

= $\simplify{{x2}+{y2}i}$

$A=\var{x2}$ and $B=\var{y2}$

Note that the real and imaginary parts of your answer should be integers (whole numbers).

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This question is used in the following exam:

Complex numbers 2024 by Katy Dobson in Katy's workspace.

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Taxonomy: mathcentre

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Michael Proudman 7 years, 8 months ago

Error in variable testing condition

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Generated value: `integer`

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DeMoivre's Theorem 1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Michael Proudman 7 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`