# How do you write the complex number in trigonometric form -9-2sqrt10i?

Oct 6, 2017

In trigonometric form expressed as $11 \left(\cos 3.754 + i \sin 3.754\right)$

#### Explanation:

$Z = a + i b$. Modulus: $| Z | = \sqrt{{a}^{2} + {b}^{2}}$;

Argument:$\theta = {\tan}^{-} 1 \left(\frac{b}{a}\right)$ Trigonometrical form : $Z = | Z | \left(\cos \theta + i \sin \theta\right)$

$Z = - 9 - 2 \sqrt{10} i$. Modulus:

$| Z | = \sqrt{{\left(- 9\right)}^{2} + {\left(- 2 \sqrt{10}\right)}^{2}} = \sqrt{81 + 40} = \sqrt{121} = 11$

Argument: $\tan \alpha = \frac{\left(| 2 \sqrt{10} |\right)}{| 9 |} = 0.7027$. $\alpha = {\tan}^{-} 1 \left(0.7027\right) = 0.61255$

Z lies on third quadrant, so $\theta = \pi + \alpha = \pi + 0.61255 \approx 3.754$

$\therefore Z = 11 \left(\cos 3.754 + i \sin 3.754\right)$, argument $\theta$ in radians

$Z = 11 \cos 3.754 + 11 \sin 3.754 i$

In trigonometric form expressed as $11 \left(\cos 3.754 + i \sin 3.754\right)$[Ans]