Question

$1 + i \sqrt{3}$ modulus and amplitude

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Answer 1

Question →

Modulus and amplitude of 1 + i√3 .

Solution →

Modulus →

Let z = 1 + i√3

So, as we know that

$|z| = \sqrt{re( {z)}^{2} + im( {z)}^{2} }$

Here Re(z) = 1 and Im(z) = √3

$|z| = \sqrt{( {1}^{2} ) + ( { \sqrt{3)} }^{2} }$

$|z| = \sqrt{4} = 2$

Hence Modulus of 1+i√3 is 2 .

Argument (amplitude) →

Let alpha be acute angle given by

$\tan(\alpha ) = | \frac{im(z)}{re(z)} |$

$\tan( \alpha ) = | \frac{ \sqrt{3} }{1} |$

$\tan( \alpha ) = \sqrt{3}$

And we know that √3 = tan 60° ,so

$\tan( \alpha ) = \tan(60) = \frac{\pi}{3}$

And here Re(z) >0 and Im(z) >0 . So it means that z lies in 1st quadrant .

ARG.Z = π/3 .

$<font color=red><marquee direction="Right">Hope it help you$

Answer 2

A number of the form a + ib , where a and b are real numbers , is defined to be a complex number

For the complex number z = a + ib , a is called real part , denoted by $\sf Re_{z}$ and b is called the imaginary part denoted by $\sf Im_{z}$ of the complex number z

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Now , let's go back to your question

$\sf \pink{ \underline{For \: modulus : \: }}$

Given ,

$\star$Complex number = 1 + i√3

$\star$a = 1

$\star$b = √3

$\sf \fbox{ \fbox{ \: Modulus \: ( |z| ) = \sqrt{ {(a)}^{2} + {(b)}^{2} } \: \: }}$

$\sf = \sqrt{ {(1)}^{2} + {( \sqrt{3} )}^{2} } \\ \\ \sf = \sqrt{1 + 3} \\ \\ \sf = \sqrt{4} \\ \\ \sf = 2$

$\sf \pink{ \underline{For \: amplitude : \: }}$

Given ,

$\star \: \: \sf r \: (Cos \: \theta) = 1 \: - - - - - \: (i) \\ \star \: \: \sf r \: (Sin \: \theta) = \sqrt{3} \: - - - - - \: (ii)$

Adding equation (i) and (ii) , we get

$\sf \implies r \: (Cos \: \theta) + r \: (Sin \: \theta) = 1 + \sqrt{3} \\ \\\sf \implies r \: (Cos \: \theta + Sin \: \theta) = 1 + \sqrt{3}$

On squaring both sides , we obtain

$\sf \implies {(r)}^{2} ( {Cos}^{2} \: \theta + {Sin }^{2} \: \theta) = {(1)}^{2} + {(\sqrt{3} )}^{2} \\ \\\sf \implies {(r)}^{2} \times 1 = 1 + 3 \\ \\\sf \implies {(r)}^{2} = 4 \\ \\\sf \implies r = \sqrt{4} \\ \\\sf \implies r = 2$

Put the value of r = 2 in equation (ii)

$\sf \implies 2(Sin \: \theta) = \sqrt{3} \\ \\\sf \implies Sin \: \theta = \frac{ \sqrt{3} }{2} \\ \\\sf \implies Sin \: \theta = Sin \: (\frac{\pi}{3} ) \\ \\\sf \implies \theta = \frac{\pi}{3}$

Therefore , the modulus and amplitude of given complex number are 2 and π/3