De Moivre's Theorem - Square Root - Numbas at mathcentre.ac.uk

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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 find a square root of $Z=\var{x}+\var{y}i$

Give any introductory information the student needs.

			Name	Type	Generated Value

x	integer	9
y	integer	6
n	number	0.5
theta	decimal	dec("5.880026035475675512456110806250854276017e-1")
mod	number	10.8166538264
x2	number	3.1477494998
y2	number	0.9530618622

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Generated value: `integer`

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

Calculate $A$ and $B$

$A$ = Gap 0.

$B$ = Gap 1.

Advice

To evaluate $\sqrt{\var{x}+\var{y}i}$ we must first express $Z=\var{x}+\var{y}i$ in polar form.

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

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 $B=\var{y2}$

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De Moivre's Theorem - Square Root

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

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Checkpoint description

Michael Proudman 7 years, 8 months ago

Error in variable testing condition

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Generated value: `integer`

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Error

De Moivre's Theorem - Square Root

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`