Sure, let's break this down into a step-by-step solution.
Given the expression:
[tex]\[
\frac{12\left(\cos 220^{\circ}+i \sin 220^{\circ}\right)}{3\left(\cos 130^{\circ}+i \sin 130^{\circ}\right)}
\][/tex]
we need to simplify it.
### Step 1: Simplify the expression
First, let's simplify the coefficients outside the trigonometric terms:
[tex]\[
\frac{12}{3} = 4
\][/tex]
So, the expression becomes:
[tex]\[
4 \cdot \frac{\left(\cos 220^{\circ} + i \sin 220^{\circ}\right)}{\left(\cos 130^{\circ} + i \sin 130^{\circ}\right)}
\][/tex]
### Step 2: Write in exponential form
We know that [tex]\( \cos \theta + i \sin \theta \)[/tex] can be written as [tex]\( e^{i\theta} \)[/tex]. Hence, we can write:
[tex]\[
\cos 220^{\circ} + i \sin 220^{\circ} = e^{i \cdot 220^{\circ}}
\][/tex]
[tex]\[
\cos 130^{\circ} + i \sin 130^{\circ} = e^{i \cdot 130^{\circ}}
\][/tex]
### Step 3: Use properties of exponents
When dividing complex exponentials, we subtract the exponents:
[tex]\[
\frac{e^{i \cdot 220^{\circ}}}{e^{i \cdot 130^{\circ}}} = e^{i(220^{\circ} - 130^{\circ})} = e^{i \cdot 90^{\circ}}
\][/tex]
So now our expression is:
[tex]\[
4 \cdot e^{i \cdot 90^{\circ}}
\][/tex]
### Step 4: Convert back to trigonometric form
We know that:
[tex]\[
e^{i \cdot 90^{\circ}} = \cos 90^{\circ} + i \sin 90^{\circ}
\][/tex]
We substitute the values:
[tex]\[
\cos 90^{\circ} = 0 \quad \text{and} \quad \sin 90^{\circ} = 1
\][/tex]
Thus,
[tex]\[
e^{i \cdot 90^{\circ}} = 0 + i \cdot 1 = i
\][/tex]
### Step 5: Multiply by 4
Finally, multiply by 4:
[tex]\[
4 \cdot i = 4i
\][/tex]
So the simplified form of the given expression is:
[tex]\[
\boxed{0 + 4i}
\][/tex]
Equivalently, the final answer is:
[tex]\[
0 + 4i \quad \text{or simply} \quad 4i
\][/tex]
From this detailed process, the step-by-step answer to the initial problem is [tex]\( \boxed{0 + 4i} \)[/tex] or just [tex]\( \boxed{4i} \)[/tex].
