Question

FIND AMPLITUDE OF THIS


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Answer 1

Step-by-step explanation:

Given:

$z = \frac{1 - \sqrt{3}i }{ \sqrt{3 + i} }$

To find: Amplitude of Complex number

Solution:

Find the square root of

$\sqrt{3 + i}$

Let

$\sqrt{3 +i} = x + iy \\ \\ square \: both \: sides \\ \\ 3 +i = {x}^{2} - {y}^{2} + 2ixy \\ \\ \text{compare real \: term \: and \: imaginary \: term} \\ \\ {x}^{2} - {y}^{2} = 3 \: \: \: ...eq1 \\ \\ 2xy = 1 \\ \\ \text{take \: modulus \: of \: complex \: number} \\ \\ {x}^{2} + {y}^{2} = 10...eq2$

From eq1 and eq2

${x}^{2} - {y}^{2} = 3 \\ {x}^{2} + {y}^{2} = 10 \\ - - - - - - \\ 2 {x}^{2} = 13 \\ \\ {x}^{2} = \frac{13}{2} \\ \\ {x}^{2} = 6.5 \\ \\ x = ± \sqrt{6.5}$

put the value of x in eq1

${x}^{2} - {y}^{2} = 3 \\ \\ 6.5 - {y}^{2} = 3 \\ \\ - {y}^{2} = 3 - 6.5 \\ \\ {y}^{2} = 3.5 \\ \\ y = ± \sqrt{3.5}$

Thus

$\sqrt{3 + i} = ± ( \sqrt{6.5} + i \sqrt{3.5} ) \\ \\ take \\ \\ \sqrt{3 +i} = ( \sqrt{6.5} + i \sqrt{3.5} )$

$\frac{1 - \sqrt{3}i }{ ( \sqrt{6.5} + i \sqrt{3.5} ) } \times \frac{ ( \sqrt{6.5} - i \sqrt{3.5} ) }{ ( \sqrt{6.5} - i \sqrt{3.5} ) } \\ \\ = \frac{(1 - \sqrt{3}i ) \times ( \sqrt{6.5} - i \sqrt{3.5} ) }{ ( \sqrt{6.5})^{2} + (\sqrt{3.5}) ^{2} } \\ \\ = \frac{(1 - \sqrt{3}i ) \times ( \sqrt{6.5} - i \sqrt{3.5} )}{10} \\ \\ \because {i}^{2} = -1 \\ \\ \frac{ \sqrt{6.5} - i \sqrt{3.5} - i \sqrt{19.5} + \sqrt{10.5} }{ 10 } \\ \\ = \frac{ \sqrt{6.5} + \sqrt{10.5} }{10} - i \frac{ \sqrt{3.5} + \sqrt{19.5} }{10} \\ \\ = \frac{2.55 + 3.24 }{10} - i \frac{1.87 +4.42}{10} \\ \\ \\ = 0.58 - i0.63$

Magnitude of z

$|z| = \sqrt{ {(0.58)}^{2} + ( { - 0.63)}^{2} } \\ \\ = \sqrt{0.3364 + 0.3956} \\ \\ = \sqrt{0.732} \\ \\ |z| = 0.855$

Hope it helps you.

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