Question

express 1+i✓3 in the modulus amplitude form.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given complex number is

$\rm :\longmapsto\:1 + \sqrt{3} \: i$

We know,

Modulus amplitude form is

$\rm :\longmapsto\:z = r(cos\theta + i \: sin\theta )$

where,.

• r is Modulus of complex number

• θ is argument of complex number

Let assume that

$\rm :\longmapsto\:1 + i \sqrt{3} = r(cos\theta + i \: sin\theta ) - - - (1)$

$\rm :\longmapsto\:1 + i \sqrt{3} = rcos\theta + i \: rsin\theta$

On comparing we get

$\rm :\longmapsto\:rcos\theta = 1 - - - (2)$

and

$\rm :\longmapsto\:rsin\theta = \sqrt{3} - - - (3)$

On squaring equation (2) and (3), we get

$\rm :\longmapsto\: {r}^{2} {cos}^{2} \theta = 1 - - - (4)$

and

$\rm :\longmapsto\: {r}^{2} {sin}^{2} \theta = 3 - - - (5)$

On adding equation (4) and (5), we get

$\rm :\longmapsto\: {r}^{2} {cos}^{2}\theta + {r}^{2} {sin}^{2} \theta = 1 + 3$

$\rm :\longmapsto\: {r}^{2} ({cos}^{2}\theta +{sin}^{2} \theta ) = 4$

$\rm :\longmapsto\: {r}^{2} = 4$

$\bf\implies \:r = 2 \: \: \: as \: r > 0$

On substituting the value of r in equation (2) and (3),

$\rm :\longmapsto\:cosθ = \dfrac{1}{2}$

and

$\rm :\longmapsto\:sinθ = \dfrac{ \sqrt{3} }{2}$

As sinθ > 0 and cosθ > 0

$\bf\implies \:\theta \: lies \: in \: {1}^{st} \: quadrant$

$\bf\implies \:\theta = \dfrac{\pi}{3}$

On substituting the value of r and θ in equation (1), we get

$\bf :\longmapsto\:1 + i \sqrt{3} = 2 \bigg(cos \dfrac{\pi}{3} + i \: sin\dfrac{\pi}{3} \bigg )$

Additional Information :-

The argument of complex number is

$\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c|c}\sf Re(z)&\sf \: Im(z)&\sf \:arg(z)\\\frac{\qquad \qquad}{}&\frac{\qquad \qquad}{}&\frac{\qquad \qquad}{}\\\sf > 0&\sf > 0&\sf\theta \\\\\sf < 0&\sf > 0&\sf\pi - \theta \\\\\sf < 0 &\sf < 0&\sf\ - \pi + \theta \\\\\sf > 0&\sf < 0&\sf - \theta \\\\ \frac{\qquad}{}&\frac{\qquad}{}&\frac{\qquad \qquad}{}\\\sf & \sf & \end{array}}\end{gathered}\end{gathered}$