AI Tutor
What happens geometrically when you multiply two complex numbers? Instead of standard algebraic gridlock, the lengths multiply while the angles simply add together.
In modulus-argument form (or polar form), any complex number is written as z=r(cos⁑heta+isin⁑heta)z = r(\cos heta + i\sin heta), where the modulus r=∣z∣r = |z| is the distance from the origin and the argument heta=arg⁑(z) heta = \arg(z) is the counterclockwise angle from the positive real axis.
Why do the angles combine so neatly under standard multiplication? The secret lies in trigonometric addition identities.

Deriving the Polar Product Rule

In 1748, mathematician Leonhard Euler formalized how trigonometry and complex numbers unite by expanding two multiplied polar expressions.
Let z1=r1(cos⁑heta1+isin⁑heta1)z_1 = r_1(\cos heta_1 + i\sin heta_1) and z2=r2(cos⁑heta2+isin⁑heta2)z_2 = r_2(\cos heta_2 + i\sin heta_2). Multiplying them with FOIL and using i2=βˆ’1i^2 = -1 gives:
z1z2=r1r2[(cos⁑heta1cos⁑heta2βˆ’sin⁑heta1sin⁑heta2)+i(sin⁑heta1cos⁑heta2+cos⁑heta1sin⁑heta2)]z_1 z_2 = r_1 r_2 [(\cos heta_1 \cos heta_2 - \sin heta_1 \sin heta_2) + i(\sin heta_1 \cos heta_2 + \cos heta_1 \sin heta_2)]
The compound angle formulae state that cos⁑(A+B)=cos⁑Acos⁑Bβˆ’sin⁑Asin⁑B\cos(A+B) = \cos A\cos B - \sin A\sin B and sin⁑(A+B)=sin⁑Acos⁑B+cos⁑Asin⁑B\sin(A+B) = \sin A\cos B + \cos A\sin B.
Substituting these identities collapses the entire product into one concise statement: ∣z1z2∣=r1r2|z_1 z_2| = r_1 r_2 and arg⁑(z1z2)=heta1+heta2\arg(z_1 z_2) = heta_1 + heta_2.
How can we use this rule to turn messy, non-standard angles into simple exact values?

Worked Example: Exact Value Simplification