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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+ in
ii) Express complex number in the polar form

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 $\mathrm{a}+$ in $\newline$ ii) Express complex number in the polar form

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Convert complex numbers between rectangular and polar form

Full solution

Q. 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

\mathrm{a}+

in

\newline

ii) Express complex number in the polar form

Rewrite in Rectangular Form: Rewrite the complex number in rectangular form by rationalizing the denominator. $\newline$ $z = \frac{5 - \sqrt{3}i}{4 + 2\sqrt{3}i} \times \frac{4 - 2\sqrt{3}i}{4 - 2\sqrt{3}i}$
Perform Multiplication: Perform the multiplication in the numerator and the denominator. $\newline$ Numerator: $(5 - \sqrt{3}i)(4 - 2\sqrt{3}i) = 20 - 10\sqrt{3}i - 4\sqrt{3}i + 6 = 26 - 14\sqrt{3}i$ $\newline$ Denominator: $(4 + 2\sqrt{3}i)(4 - 2\sqrt{3}i) = 16 - 12 = 4$
Simplify by Division: Divide the numerator by the denominator to simplify. $\newline$ $z = \frac{26 - 14\sqrt{3}i}{4} = 6.5 - 3.5\sqrt{3}i$
Calculate Magnitude: Convert the rectangular form $z = 6.5 - 3.5\sqrt{3}i$ to polar form. $\newline$ Calculate the magnitude $r$ . $\newline$ $r = \sqrt{(6.5)^2 + (3.5\sqrt{3})^2} = \sqrt{42.25 + 36.75} = \sqrt{79}$
Calculate Angle: Calculate the angle $\theta$ . $\newline$ $\theta = \tan^{-1}\left(\frac{-3.5\sqrt{3}}{6.5}\right)$
Express in Polar Form: Express in polar form using $r$ and $\theta$ . $\newline$ $z = 79(\cos(\theta) + i\sin(\theta))$

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