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Geometrical Representation of a Complex Number

Argument of z...

Question

Argument of $z$ if $z = sin (\frac{π}{5}) + i (1 - cos (\frac{π}{5}))$ is

A

\frac{π}{5}

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B

\frac{π}{10}

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C

\frac{π}{15}

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D

\frac{π}{20}

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Solution

The correct option is B $\frac{π}{10}$
We have $z = sin (\frac{π}{5}) + i (1 - cos (\frac{π}{5}))$
$= 2 sin (\frac{π}{10}) cos (\frac{π}{10}) + i \cdot 2 {sin}^{2} (\frac{π}{10})$
$= 2 sin (\frac{π}{10}) [cos (\frac{π}{10}) + i sin (\frac{π}{10})]$

which is of form $z = | z | (cos θ + i sin θ)$
$∴ arg (z) = \frac{π}{10}, | z | = 2 sin (\frac{π}{10})$

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The amplitude of
$s i n \frac{π}{5} + i (1 - c o s \frac{π}{5})$

Express $s i n \frac{π}{5} + i (1 - c o s \frac{π}{5})$ in polar form.

Express the complex number $s i n \frac{π}{5} + i (1 - c o s \frac{π}{5})$ in polar form.

{⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{sin \frac{π}{6} + i (1 - cos \frac{π}{6})}{sin \frac{π}{6} - i (1 - cos \frac{π}{6})} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦}^{2}

i = \sqrt{- 1}

z = 1 + c o s (\frac{π}{5}) + i s i n (\frac{π}{5}),

then {sin (arg(z))} is equal to

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Geometrical Representation of a Complex Number

Standard XII Mathematics

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