Question

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

Answer 1

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

Explanation:

Multiplication of Complex Numbers:

How do we multiply two complex numbers?

Assume that we want to multiply two complex numbers:

`color(green)((a+bi)*(c+di))`

We can use the FOIL method to multiply:

Firsts: `a * c`

Outers: `a * di`

Inners: `bi * c`

Lasts: `bi * di`

Hence,

`color(green)((a+bi)*(c+di)=ac+adi+bci+bdi^2)`

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

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

`rArr [1 * (-2)] + [1 (-3i)] + [(-2i) (-2)] + [(-2i) * (-3i)]`

`rArr (-2) + (-3i) + (+4i) + (+6i^2)`

`rArr (-2) + (+1i) + (6i^2)`

Note that `color(red)(" "i^2 = -1)`

Hence,

`rArr (-2) + (+1i) + [6*(-1)]`

`rArr (-2) + (+1i) + (-6)`

`rArr (-8 +i)`

Hence, our intermediate answer:

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

We also know that, in trigonometric form of complex numbers

`Z = x + iy`

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

`x = r Cos (Theta)`

`y = r Sin (Theta)`

`Z = r[Cos(Theta) + i Sin(Theta)]`

We will now calculatre

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

`rArr |Z| = sqrt(1^2 + (-8)^2)`

`rArr |Z| = sqrt(65)`

`Theta = arctan(-1/8)`

`Theta = -3.2659`

We have, `Z = r[Cos(Theta) + i Sin(Theta)]`

`Z = sqrt(65)[Cos(-3.2659) + i Sin(-3.2659)]`

Hope this helps.