To find the quotient [tex]\(\frac{z_1}{z_2}\)[/tex] where [tex]\(z_1 = 21(\cos 200^\circ + i \sin 200^\circ)\)[/tex] and [tex]\(z_2 = 7(\cos 120^\circ + i \sin 120^\circ)\)[/tex], we can use the property of division of complex numbers in polar form.
1. Determine the magnitude of [tex]\(\frac{z_1}{z_2}\)[/tex]:
The magnitude (or modulus) of a complex number in polar form [tex]\(r(\cos \theta + i \sin \theta)\)[/tex] is [tex]\(r\)[/tex]. So the magnitudes of [tex]\(z_1\)[/tex] and [tex]\(z_2\)[/tex] are:
[tex]\[
|z_1| = 21 \quad \text{and} \quad |z_2| = 7
\][/tex]
The magnitude of the quotient [tex]\(\frac{z_1}{z_2}\)[/tex] is:
[tex]\[
\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|} = \frac{21}{7} = 3
\][/tex]
2. Determine the argument of [tex]\(\frac{z_1}{z_2}\)[/tex]:
The argument (or angle) of a complex number [tex]\(r(\cos \theta + i \sin \theta)\)[/tex] is [tex]\(\theta\)[/tex]. So the arguments of [tex]\(z_1\)[/tex] and [tex]\(z_2\)[/tex] are:
[tex]\[
\arg(z_1) = 200^\circ \quad \text{and} \quad \arg(z_2) = 120^\circ
\][/tex]
The argument of the quotient [tex]\(\frac{z_1}{z_2}\)[/tex] is:
[tex]\[
\arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2) = 200^\circ - 120^\circ = 80^\circ
\][/tex]
3. Combine the results:
Combine the magnitude and the argument to express the quotient in polar form:
[tex]\[
\frac{z_1}{z_2} = 3\left(\cos 80^\circ + i \sin 80^\circ\right)
\][/tex]
Therefore, the correct answer is:
[tex]\[
\frac{z_1}{z_2} = 3\left(\cos 80^\circ + i \sin 80^\circ\right)
\][/tex]
