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COMPLEX NUMBERS AND QUADRATIC EQUATIONS — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSCOMPLEX NUMBERS AND QUADRATIC EQUATIONS2 Marks
Question
Convert the complex number in the polar form: 1 - i
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Answer

Here $z = 1 - i = r(\cos \theta + i\sin \theta )$
$\ \Rightarrow r\cos \theta = 1$ and $r\sin \theta = - 1$ .... (i)
Squaring both sides of (i) and adding
${r^2}({\cos ^2}\theta + {\sin ^2}\theta ) = 1 + 1$$ \Rightarrow {r^2} = 2 \Rightarrow r = \sqrt 2 $
$\therefore \sqrt 2 \cos \theta = 1$ and $\sqrt 2 \sin \theta = - 1$
$ \Rightarrow \cos \theta = \frac{1}{{\sqrt 2 }}$ and $\sin \theta = \frac{{ - 1}}{{\sqrt 2 }}$
Since $\sin \theta $ is negative and $\cos \theta $ is positive
$\therefore \theta $ lies in fourth quadrant
$\therefore \theta = \frac{{ - \pi }}{4}$
Hence polar form of z is $\sqrt 2 \left[ {\cos \left( {\frac{{ - \pi }}{4}} \right) + i\sin \left( {\frac{{ - \pi }}{4}} \right)} \right]$

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