Multiplication and division in polar form (WACE Year 12 Mathematics Specialist Unit 3)

What this dot point is asking

SCSA wants you to combine complex numbers in modulus-argument form and read off the geometric meaning: multiplication as a rotation plus dilation about the origin.

The multiplication rule

Write $z_1 = r_1\,\text{cis}\,\theta_1$ and $z_2 = r_2\,\text{cis}\,\theta_2$. Expanding the product and using the angle-sum identities $\cos(\theta_1 + \theta_2)$ and $\sin(\theta_1 + \theta_2)$ gives

So $|z_1 z_2| = |z_1|\,|z_2|$ and $\arg(z_1 z_2) = \arg z_1 + \arg z_2$ (up to a multiple of $2\pi$, since the result may need adjusting back into the principal range).

The division rule

Dividing reverses the process:

Moduli divide and arguments subtract. As a special case $\tfrac{1}{z} = \tfrac{1}{r}\,\text{cis}(-\theta)$, the reciprocal having reciprocal modulus and negated argument.

Geometric interpretation

Two clean cases worth memorising: multiplying by $i = \text{cis}\,\tfrac{\pi}{2}$ rotates by a quarter turn anticlockwise with no scaling, and multiplying by $-1 = \text{cis}\,\pi$ rotates by a half turn. The conjugate in polar form is $\overline{r\,\text{cis}\,\theta} = r\,\text{cis}(-\theta)$, a reflection in the real axis.

Deriving the rule from the angle-sum identities

The product rule is not a definition to memorise but a consequence of the trigonometric angle-sum identities. Multiplying $r_1(\cos\theta_1 + i\sin\theta_1)$ by $r_2(\cos\theta_2 + i\sin\theta_2)$ and expanding gives a real part $r_1 r_2(\cos\theta_1\cos\theta_2 - \sin\theta_1\sin\theta_2)$ and an imaginary part $r_1 r_2(\sin\theta_1\cos\theta_2 + \cos\theta_1\sin\theta_2)$. The bracketed expressions are exactly $\cos(\theta_1 + \theta_2)$ and $\sin(\theta_1 + \theta_2)$, so the product is $r_1 r_2\,\text{cis}(\theta_1 + \theta_2)$. Knowing this derivation is worth marks when a question asks you to justify the rule rather than just apply it.

Transformations of the plane

Because multiplication by $r\,\text{cis}\,\alpha$ rotates and scales, complex multiplication is a compact way to describe geometric transformations about the origin. A rotation by $\alpha$ alone is multiplication by $\text{cis}\,\alpha$ (modulus $1$); a dilation by factor $k$ alone is multiplication by the real number $k$; and a spiral similarity combining both is multiplication by $k\,\text{cis}\,\alpha$. This viewpoint connects the complex-numbers topic to the geometry of the Argand plane and to the later study of curves and regions.

Why polar form helps

Cartesian multiplication is algebraically heavy, but polar form reduces it to one product and one sum. This is also the engine behind de Moivre's theorem, which is just repeated multiplication of equal factors.