# How do you multiply  (1-2i)(-2-3i)  in trigonometric form?

Dec 23, 2017

$\textcolor{b l u e}{\left(1 - 2 i\right) \left(- 2 - 3 i\right) = - 8 + i}$

#### Explanation:

Multiplication of Complex Numbers:

How do we multiply two complex numbers?

Assume that we want to multiply two complex numbers:

$\textcolor{g r e e n}{\left(a + b i\right) \cdot \left(c + \mathrm{di}\right)}$

We can use the FOIL method to multiply:

Firsts: $a \cdot c$

Outers: $a \cdot \mathrm{di}$

Inners: $b i \cdot c$

Lasts: $b i \cdot \mathrm{di}$

Hence,

$\textcolor{g r e e n}{\left(a + b i\right) \cdot \left(c + \mathrm{di}\right) = a c + a \mathrm{di} + b c i + b {\mathrm{di}}^{2}}$

Now, we will consider our problem and multiply the complex numbers given to us:

color(blue)((1-2i)(-2-3i)

$\Rightarrow \left[1 \cdot \left(- 2\right)\right] + \left[1 \left(- 3 i\right)\right] + \left[\left(- 2 i\right) \left(- 2\right)\right] + \left[\left(- 2 i\right) \cdot \left(- 3 i\right)\right]$

$\Rightarrow \left(- 2\right) + \left(- 3 i\right) + \left(+ 4 i\right) + \left(+ 6 {i}^{2}\right)$

$\Rightarrow \left(- 2\right) + \left(+ 1 i\right) + \left(6 {i}^{2}\right)$

Note that $\textcolor{red}{\text{ } {i}^{2} = - 1}$

Hence,

$\Rightarrow \left(- 2\right) + \left(+ 1 i\right) + \left[6 \cdot \left(- 1\right)\right]$

$\Rightarrow \left(- 2\right) + \left(+ 1 i\right) + \left(- 6\right)$

$\Rightarrow \left(- 8 + i\right)$

color(blue)((1-2i)(-2-3i)=-8 +i

We also know that, in trigonometric form of complex numbers

$Z = x + i y$

r = |Z| = sqrt(x ^ 2 + y ^2

$x = r C o s \left(\Theta\right)$

$y = r S \in \left(\Theta\right)$

$Z = r \left[C o s \left(\Theta\right) + i S \in \left(\Theta\right)\right]$

We will now calculatre

r = |Z| = sqrt(x ^ 2 + y ^2

$\Rightarrow | Z | = \sqrt{{1}^{2} + {\left(- 8\right)}^{2}}$

$\Rightarrow | Z | = \sqrt{65}$

$\Theta = \arctan \left(- \frac{1}{8}\right)$

$\Theta = - 3.2659$

We have, $Z = r \left[C o s \left(\Theta\right) + i S \in \left(\Theta\right)\right]$

$Z = \sqrt{65} \left[C o s \left(- 3.2659\right) + i S \in \left(- 3.2659\right)\right]$

Hope this helps.