Question

amplitude complex number of -√3+i is

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Answer 1

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

Given complex number is

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

Let we assume that

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

can be rewritten as

$\rm :\longmapsto\: - \sqrt{3} + i = rcos\theta + i \: r \: sin\theta$

On comparing Real and Imaginary parts, we have

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

and

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

On squaring equation (1) and (2) and adding, we get

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

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

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

$\bf\implies \:r = 2$

On substituting the value of r in equation (1) and (2), we get

$\rm :\longmapsto\:cos\theta = - \dfrac{ \sqrt{3} }{2}$

and

$\rm :\longmapsto\:sin\theta = \dfrac{1}{2}$

$\bf\implies \:\theta \: lies \: in \: {2}^{nd} \: quadrant$

$\bf\implies \:\theta = \pi - \dfrac{\pi}{6}$

$\bf\implies \:\theta = \dfrac{5\pi}{6}$

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Additional Information

Short cut trick to find amplitude or argument of complex number

$\begin{gathered}\boxed{\begin{array}{c|c} \bf Complex \: number & \bf arg(z) \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \1sf x + iy & \sf  {tan}^{ - 1}\bigg |\dfrac{y}{x} \bigg|   \\ \\ \sf  - x + iy & \sf \pi - {tan}^{ - 1}\bigg |\dfrac{y}{x}\bigg | \\ \\ \sf  - x - iy & \sf  - \pi + {tan}^{ - 1}\bigg |\dfrac{y}{x}\bigg | \\ \\ \sf x - iy & \sf  - {tan}^{ - 1}\bigg |\dfrac{y}{x}\bigg | \end{array}} \\ \end{gathered}$

Answer 2

Step-by-step explanation:

$\mathtt{ \implies - \sqrt{3} + i}$

$\mathtt{ \implies \frac{1}{2}[ - \frac{ \sqrt{3} }{2} + i \frac{1}{2} ] }$

$\mathtt{ \implies \frac{1}{2}[ \cos( \frac{5\pi}{6} ) + i \sin(x) \frac{5\pi}{6} ] }$

$\mathtt{ \implies \: amp( - \sqrt{3} + i)} = \frac{5\pi}{6}$

hope it help you.