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Properties of Argument

Find the modu...

Question

Find the modulus, argument and the principal argument of the complex numbers.
$z = 1 + c o s \frac{10 π}{9} + i s i n (\frac{10 π}{9})$

Principal Arg

z = - \frac{4 π}{9}; | z | = 2 c o s \frac{4 π}{9}; A r g z = 2 k π - \frac{4 π}{9} k ϵ l

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Principal Arg

z = - \frac{10 π}{9}; | z | = 2 c o s \frac{10 π}{9}; A r g z = 2 k π - \frac{10 π}{9} k ϵ l

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Principal Arg

z = - \frac{- 10 π}{9}; | z | = 2 c o s \frac{- 10 π}{9}; A r g z = 2 k π - \frac{4 π}{9} k ϵ l

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Principal Arg

z = - \frac{- 4 π}{9}; | z | = 2 c o s \frac{- 4 π}{9}; A r g z = 2 k π - \frac{4 π}{9} k ϵ l

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Solution

The correct option is A Principal Arg $z = - \frac{4 π}{9}; | z | = 2 c o s \frac{4 π}{9}; A r g z = 2 k π - \frac{4 π}{9} k ϵ l$
Simplifying, we get
$z = 2 {cos}^{2} (\frac{5 π}{9}) + 2 i sin (\frac{5 π}{9}) cos (\frac{5 π}{9})$
$= 2 cos (\frac{5 π}{9}) [cos (\frac{5 π}{9}) + i sin (\frac{5 π}{9})]$
$= - 2 cos (\frac{4 π}{9}) [- cos (\frac{4 π}{9}) + i sin (\frac{4 π}{9})]$
$= 2 cos (\frac{4 π}{9}) [cos (\frac{4 π}{9}) - i sin (\frac{4 π}{9})]$
Hence
$| z | = 2 cos (\frac{4 π}{9})$
And
Principal Arg $(z) = - \frac{4 π}{9}$

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