# How do you divide ( 4i+4) / (6i +5 ) in trigonometric form?

Feb 27, 2018

In trigonometric form: $0.725 \left(\cos 0.091 - i \sin 0.091\right)$

#### Explanation:

 (4+4i)/(5+6i) ;Z=a+ib . Modulus: $| Z | = \sqrt{{a}^{2} + {b}^{2}}$;

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

$Z = | Z | \left(\cos \theta + i \sin \theta\right)$ ${Z}_{1} = 4 + 4 i$

Modulus:$| {Z}_{1} | = \sqrt{{4}^{2} + {4}^{2}} \approx 5.66$

Argument: $\tan \alpha = \frac{| 4 |}{| 4 |} \therefore \alpha = {\tan}^{-} 1 \left(1\right) = 0.785$

${Z}_{1}$ lies on first quadrant, so $\theta = \alpha \approx 0.785$

$\therefore {Z}_{1} = 5.66 \left(\cos 0.785 + i \sin 0.785\right)$

${Z}_{2} = 5 + 6 i$. Modulus:$| Z | = \sqrt{{5}^{2} + {6}^{2}}$

$= \sqrt{61} \approx 7.81$ Argument: $\tan \alpha = \frac{| 6 |}{| 5 |}$

=6/5 :.alpha =tan^-1 (1.2) ~~ 0.0876 ; Z_2 lies on first

quadrant.$\therefore \theta = \alpha \approx 0.876$

$\therefore {Z}_{2} = 7.81 \left(\cos 0.876 + i \sin 0.876\right) \therefore \frac{4 + 4 i}{5 + 6 i} =$

 Z= (5.66(cos0.785+isin 0.785))/(7.81(cos 0.876+isin 0.876)

$Z = 0.725 \left(\cos \left(0.785 - 0.876\right) + i \sin \left(0.785 - 0.876\right)\right)$ or

$Z = 0.725 \left(\cos \left(- 0.091\right) + i \sin \left(- 0.091\right)\right)$ or

$Z = 0.725 \left(\cos \left(0.091\right) - i \sin \left(0.091\right)\right) = \left(\frac{44}{61} - \frac{4}{61} i\right)$

In trigonometric form: $0.725 \left(\cos 0.091 - i \sin 0.091\right)$ [Ans]