# How do you divide  (1-2i) / (6+i)  in trigonometric form?

Jan 30, 2018

In trigonometric form: $0.368 \left(\cos 5.011 + i \sin 5.011\right)$

#### Explanation:

$Z = \frac{1 - 2 i}{6 + i}$

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

Argument:$\theta = {\tan}^{-} 1 \left(\frac{b}{a}\right)$ Trigonometrical form :

$Z = | Z | \left(\cos \theta + i \sin \theta\right)$

${Z}_{1} = 1 - 2 i$.Modulus:$| Z | = \sqrt{{1}^{2} + {\left(- 2\right)}^{2}}$

$= \sqrt{5} \approx 2.236$ Argument: $\tan \alpha = \frac{| - 2 |}{| 1 |}$

$= 2$. alpha =tan^-1 2 = 1.107; Z_1 lies on fourth quadrant, so

$\theta = 2 \pi - \alpha = 2 \pi - 1.107 \approx 5.176$

$\therefore {Z}_{1} = 2.236 \left(\cos 5.176 + i \sin 5.176\right)$,

${Z}_{2} = 6 + i$.Modulus:$| Z | = \sqrt{{6}^{2} + {1}^{2}}$

$= \sqrt{37} \approx 6.083$ Argument: $\tan \alpha = \frac{| 1 |}{| 6 |}$

=1/6 :.alpha =tan^-1 (1/6) = 0.165 ; Z_2 lies on first quadrant,

$\therefore \theta = \alpha \approx 0.165$ $\therefore {Z}_{2} = 6.083 \left(\cos 0.165 + i \sin 0.165\right)$

$Z = \frac{1 - 2 i}{6 + i}$

 Z= (2.236(cos5.176+isin 5.176))/(6.083(cos 0.165+isin 0.165)

$Z = 0.368 \left(\cos \left(5.176 - 0.165\right) + i \sin \left(5.176 - 0.165\right)\right)$ or

$Z = 0.368 \left(\cos 5.011 + i \sin 5.011\right) = \frac{4}{37} - \frac{13}{37} i$

In trigonometric form: $0.368 \left(\cos 5.011 + i \sin 5.011\right)$ [Ans]