# How do you write the complex number in trigonometric form 5/2(sqrt3-i)?

In trigonometric form expressed as $5 \left(\cos 330 + i \sin 330\right)$
$Z = a + i b$. Modulus: $| Z | = \sqrt{{a}^{2} + {b}^{2}}$; Argument:$\theta = {\tan}^{-} 1 \left(\frac{b}{a}\right)$ Trigonometrical form : $Z = | Z | \left(\cos \theta + i \sin \theta\right)$
$Z = \frac{5}{2} \left(\sqrt{3} - i\right) = \frac{5}{2} \sqrt{3} - \frac{5}{2} i$. Modulus $| Z | = \sqrt{{\left(\frac{5}{2} \sqrt{3}\right)}^{2} + {\left(- \frac{5}{2}\right)}^{2}} = \sqrt{\frac{75}{4} + \frac{25}{4}} = \sqrt{25} = 5$
Argument: $\tan \theta = \frac{- \cancel{\frac{5}{2}}}{\cancel{\frac{5}{2}} \sqrt{3}} = - \frac{1}{\sqrt{3}}$. Z lies on fourth quadrant, so $\theta = {\tan}^{-} 1 \left(- \frac{1}{\sqrt{3}}\right) = - \frac{\pi}{6} = - {30}^{0} \mathmr{and} \theta = 360 - 30 = {330}^{0} \therefore Z = 5 \left(\cos 330 + i \sin 330\right)$
In trigonometric form expressed as $5 \left(\cos 330 + i \sin 330\right)$[Ans]