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Assignment 3

Consider the complex number (MARCH-2012)

z=(5-sqrt3i)/(4+2sqrt3i)
i) Express complex number in the form of
a+ib.
ii) Express complex number in the polar form

$8$ : $52$ PM $\mid 155 \mathrm{~KB} / \mathrm{s}$ $\newline$ Edit $\newline$ Assignment $3$ $\newline$ $1$ . Consider the complex number (MARCH $-2012$ ) $\newline$ $z=\frac{5-\sqrt{3} i}{4+2 \sqrt{3} i}$ $\newline$ i) Express complex number in the form of $a+i b$ . $\newline$ ii) Express complex number in the polar form

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Convert complex numbers between rectangular and polar form

Full solution

Q.

8

:

52

PM

\mid 155 \mathrm{~KB} / \mathrm{s}

\newline

Edit

\newline

Assignment

3

\newline

1

. Consider the complex number (MARCH

-2012

)

\newline

z=\frac{5-\sqrt{3} i}{4+2 \sqrt{3} i}

\newline

i) Express complex number in the form of

a+i b

.

\newline

ii) Express complex number in the polar form

Multiply by Conjugate: To express $z$ in the form $a + ib$ , we first simplify the fraction by multiplying the numerator and the denominator by the conjugate of the denominator. $\newline$ Calculation: $z = \frac{5 - \sqrt{3}i}{4 + 2\sqrt{3}i} \times \frac{4 - 2\sqrt{3}i}{4 - 2\sqrt{3}i}$
Simplify Denominator: Simplify the denominator using the formula $(a + bi)(a - bi) = a^2 + b^2$ . $\newline$ Calculation: Denominator = $4^2 + (2\sqrt{3})^2 = 16 + 12 = 28$
Expand Numerator: Expand the numerator using the distributive property. $\newline$ Calculation: Numerator = $(5 - \sqrt{3}i)(4 - 2\sqrt{3}i) = 20 - 10\sqrt{3}i - 4\sqrt{3}i + 6 = 26 - 14\sqrt{3}i$
Divide Real and Imaginary: Divide the real and imaginary parts of the numerator by the denominator. $\newline$ Calculation: $z = \frac{26}{28} - \frac{14\sqrt{3}i}{28} = \frac{13}{14} - \frac{7\sqrt{3}i}{14}$
Calculate Magnitude: To express $z$ in polar form, calculate the magnitude $r$ and the angle $\theta$ . $\newline$ Calculation: $r = \sqrt{\left(\frac{13}{14}\right)^2 + \left(\frac{7\sqrt{3}}{14}\right)^2}$
Continue Calculating: Continue calculating the magnitude. $\newline$ Calculation: $r = \sqrt{\frac{169}{196} + \frac{147}{196}} = \sqrt{\frac{316}{196}} = \sqrt{\frac{158}{98}}$

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