To find the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] given [tex]\(z_1=2 \operatorname{cis} 120^{\circ}\)[/tex] and [tex]\(z_2=4 \operatorname{cis} 30^{\circ}\)[/tex], we need to divide these two complex numbers in polar form and then convert the result to rectangular form.
### Step-by-Step Solution
1. Divide the Moduli:
[tex]\[
\text{Modulus of } \frac{z_1}{z_2} = \frac{\text{Modulus of } z_1}{\text{Modulus of } z_2} = \frac{2}{4} = \frac{1}{2}
\][/tex]
2. Subtract the Arguments:
[tex]\[
\text{Argument of } \frac{z_1}{z_2} = \text{Argument of } z_1 - \text{Argument of } z_2 = 120^{\circ} - 30^{\circ} = 90^{\circ}
\][/tex]
3. Convert the Argument from Degrees to Radians:
[tex]\[
90^{\circ} = \frac{\pi}{2} \text{ radians}
\][/tex]
4. Calculate the Rectangular Form:
Given the polar form [tex]\(\frac{z_1}{z_2} = \frac{1}{2} \operatorname{cis} 90^{\circ}\)[/tex], we convert this to rectangular form using the cosine and sine functions:
[tex]\[
\frac{1}{2} \operatorname{cis} 90^{\circ} = \frac{1}{2} (\cos 90^{\circ} + i \sin 90^{\circ})
\][/tex]
Using the values of cosine and sine at [tex]\(90^{\circ}\)[/tex]:
[tex]\[
\cos 90^{\circ} = 0 \quad \text{and} \quad \sin 90^{\circ} = 1
\][/tex]
5. Substitute and Simplify:
[tex]\[
\frac{1}{2} (0 + i \cdot 1) = \frac{1}{2} (0 + i) = 0 + \frac{1}{2}i = 0 + 0.5i
\][/tex]
Thus, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[
a = 0
\][/tex]
[tex]\[
b = 0.5
\][/tex]
So the complete answer is:
If [tex]\(z_1=2 \operatorname{cis} 120^{\circ}, z_2=4 \operatorname{cis} 30^{\circ}\)[/tex], and [tex]\(\frac{z_1}{z_2}=a+b i\)[/tex], then [tex]\(a=0\)[/tex] and [tex]\(b=0.5\)[/tex].
