How do you divide ( i+8) / (5i +5 ) in trigonometric form?

Mar 19, 2016

${C}_{3} = \frac{40 + 5 i - 40 i + 5}{25 + 25} = \frac{45 - 35 i}{50} = \frac{1}{10} \left(9 - 7 i\right)$

Explanation:

Given the complex set ${C}_{1} = \left(i + 8\right)$ and ${C}_{2} = \left(5 i + 5\right)$
Required: $\frac{i + 8}{5 i + 5}$
Solution: Use complex conjugate of ${C}_{2}$, $\overline{{C}_{2}}$to perform complex number division.
The product of a complex number $C$ with it's conjugate $\overline{C}$ is: $R = C \cdot \overline{C} = \left(a + b i\right) \left(a - b i\right) = {a}^{2} + {b}^{2}$, a real number.
Thus multiplying top and bottom by $\overline{{C}_{2}} = 5 - 5 i$
${C}_{3} = \frac{{C}_{1} \cdot \overline{{C}_{2}}}{{C}_{2} \cdot \overline{{C}_{2}}} = \frac{\left(i + 8\right) \left(5 - 5 i\right)}{\left(5 + 5 i\right) \cdot \left(5 - 5 i\right)}$

${C}_{3} = \frac{40 + 5 i - 40 i + 5}{25 + 25} = \frac{45 - 35 i}{50} = \frac{1}{10} \left(9 - 7 i\right)$