# How do you divide ( 6i+5) / ( 7 i -4 ) in trigonometric form?

Feb 14, 2018

In trigonometric form: $0.969 \left(\cos 1.214 - i \sin 1.214\right)$

#### Explanation:

$\frac{5 + 6 i}{- 4 + 7 i}$ $Z = a + i b$. Modulus: $| Z | = \sqrt{{a}^{2} + {b}^{2}}$;

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

Z =|Z|(costheta+isintheta) ; Z= 5+6 i .

Modulus:$| Z | = \sqrt{{5}^{2} + {6}^{2}} \approx 7.81$

Argument: $\tan \alpha = \frac{| 6 |}{| 5 |} \therefore \alpha = {\tan}^{-} 1 \left(1.2\right) = 0.876$

${Z}_{1}$ lies on first quadrant, so $\theta = \alpha \approx 0.876$

$\therefore {Z}_{1} = 7.81 \left(\cos 0.876 + i \sin 0.876\right)$

${Z}_{2} = - 4 + 7 i$. Modulus:$| Z | = \sqrt{{4}^{2} + {7}^{2}}$

$= \sqrt{65} \approx 8.062$ Argument: $\tan \alpha = \frac{| 7 |}{| - 4 |}$

=7/4 :.alpha =tan^-1 (7/4) = 1.052 ; Z_2 lies on second

quadrant.$\therefore \theta = \pi - \alpha \approx 2.09$

$\therefore {Z}_{2} = 8.062 \left(\cos 2.09 + i \sin 2.09\right) \therefore \frac{5 + 6 i}{- 4 + 7 i} =$

 Z= (7.81(cos0.876+isin 0.876))/(8.062(cos 2.09+isin 2.09)

$Z = 0.969 \left(\cos \left(0.876 - 2.09\right) + i \sin \left(0.876 - 2.09\right)\right)$ or

$Z = 0.0969 \left(\cos 1.214 - i \sin 1.214\right) = \frac{22}{65} - \frac{59}{65} i$

In trigonometric form: $0.969 \left(\cos 1.214 - i \sin 1.214\right)$ [Ans]