Given two complex numbers z1=r1(cosθ1+i sinθ1) and z2=r2(cosθ2+i sinθ2), z2 ≠0, show that (z1)/(

step 1 we have problem number 158 we have given two complex numbers z1 equal to r1 cause of theta 1 plu

Given two complex numbers z1=r1(cosθ1+i sinθ1) and...

Given two complex numbers $z_{1}=r_{1}\left(\cos \theta_{1}+i \sin \theta_{1}\right)$ and $z_{2}=r_{2}\left(\cos \theta_{2}+i \sin \theta_{2}\right), z_{2} \neq 0,$ show that $$ \frac{z_{1}}{z_{2}}=\frac{r_{1}}{r_{2}}\left[\cos \left(\theta_{1}-\theta_{2}\right)+i \sin \left(\theta_{1}-\theta_{2}\right)\right] $$

Given two complex numbers $z_{1}=r_{1}\left(\cos \theta_{1}+i \sin \theta_{1}\right)$ and $z_{2}=r_{2}\left(\cos \theta_{2}+i \sin \theta_{2}\right), z_{2} \neq 0,$ show that
$$
\frac{z_{1}}{z_{2}}=\frac{r_{1}}{r_{2}}\left[\cos \left(\theta_{1}-\theta_{2}\right)+i \sin \left(\theta_{1}-\theta_{2}\right)\right]
$$

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Key Concepts

Complex Numbers in Polar Form

Complex numbers can be represented in polar form as r (cos ? + i sin ?), where r is the modulus (or absolute value) and ? is the argument (or angle). This representation is particularly useful for multiplication and division, as it allows the geometric interpretation of combining magnitudes and angles.

Modulus and Argument

The modulus, r, represents the distance of the complex number from the origin in the complex plane, and the argument, ?, is the angle measured from the positive real axis. These two components simplify operations like multiplication and division, because the magnitudes multiply or divide, and the angles add or subtract.

Division of Complex Numbers in Polar Form

When dividing two complex numbers expressed in polar form, the division is performed by dividing their moduli and subtracting their arguments. This means that given z1 = r1 (cos ?1 + i sin ?1) and z2 = r2 (cos ?2 + i sin ?2), the quotient is (r1/r2) times (cos (?1 - ?2) + i sin (?1 - ?2)), reflecting the geometric interpretation of scaling the magnitude and rotating by the difference in angles.

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