If arg z1 = π/4 and arg z2 = π/6, what is arg(z1 z2) (modulo 2π)?

If arg z1 = π/4 and arg z2 = π/6, what is arg(z1 z2) (modulo 2π)?

• A. π/2
• B. π/12
• C. 7π/12
• D. 5π/12

Answer: (D)

When multiplying two complex numbers in polar form, their arguments add and you take the result modulo 2π. If z1 = r1 e^{iπ/4} and z2 = r2 e^{iπ/6}, then z1 z2 has argument π/4 + π/6 modulo 2π. Compute π/4 + π/6 = 3π/12 + 2π/12 = 5π/12. This value is already between 0 and 2π, so no further adjustment is needed. Therefore arg(z1 z2) ≡ 5π/12 (mod 2π). This corresponds to 75°, while the other options represent 90°, 15°, and 105° respectively, which are not the sum of the given angles.

When multiplying two complex numbers in polar form, their arguments add and you take the result modulo 2π. If z1 = r1 e^{iπ/4} and z2 = r2 e^{iπ/6}, then z1 z2 has argument π/4 + π/6 modulo 2π. Compute π/4 + π/6 = 3π/12 + 2π/12 = 5π/12. This value is already between 0 and 2π, so no further adjustment is needed. Therefore arg(z1 z2) ≡ 5π/12 (mod 2π). This corresponds to 75°, while the other options represent 90°, 15°, and 105° respectively, which are not the sum of the given angles.