# How do you divide  (7-i) / (3-i)  in trigonometric form?

Aug 11, 2018

In trigonometric form: $2.236 \left(\cos 0.18 + i \sin 0.18\right)$

#### Explanation:

$Z = \frac{7 - i}{3 - i}$

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

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

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

${Z}_{1} = 7 - i$.Modulus:$| {Z}_{1} | = \sqrt{{7}^{2} + {\left(- 1\right)}^{2}}$

$= \sqrt{50} \approx 7.07$ Argument: $\tan \alpha = \frac{\left(| - 1 |\right)}{| 7 |}$

$= \frac{1}{7} , \alpha = {\tan}^{-} 1 \left(\frac{1}{7}\right) \approx 0.142 , Z$ lies on fourth quadrant,

so $\theta = 2 \pi - \alpha = 2 \pi - 0.142 \approx 6.14$

$\therefore {Z}_{1} = 7.07 \left(\cos 6.14 + i \sin 6.14\right)$,

${Z}_{2} = 3 - i$.Modulus:$| {Z}_{2} | = \sqrt{{3}^{2} + {\left(- 1\right)}^{2}}$

$= \sqrt{10} \approx 3.16$ Argument: $\tan \alpha = \frac{\left(| - 1 |\right)}{| 3 |}$

$= \frac{1}{3} , \alpha = {\tan}^{-} 1 \left(\frac{1}{3}\right) \approx 0.322 , Z$ lies on fourth quadrant,

so $\theta = 2 \pi - \alpha = 2 \pi - 0.322 \approx 5.96$

$\therefore {Z}_{2} = 3.16 \left(\cos 5.96 + i \sin 5.96\right)$,

$Z = \frac{7 - i}{3 - i}$

 Z= (7.07(cos 6.14+i sin 6.14))/(3.16(cos 5.96+i sin 5.96)

$Z = 2.236 \left(\cos \left(6.14 - 5.96\right) + i \sin \left(6.14 - 5.96\right)\right)$ or

$Z = 2.236 \left(\cos 0.18 + i \sin 0.18\right) = 2.2 + 0.4 i$

In trigonometric form; $2.236 \left(\cos 0.18 + i \sin 0.18\right)$ [Ans]