Question

If \(\Arg z_1\) and \(\Arg z_2\) are \(\frac{\pi}{3}\) and \(\frac{\pi}{5}\) respectively, then the value of \(\Arg z_1 + \Arg z_2\) is:

• A. \(\frac{11\pi}{15}\)
• B. \(\frac{6\pi}{15}\)
• C. \(\frac{2\pi}{15}\)
• D. \(\frac{8\pi}{15}\)

Hint:

When summing arguments, remember the principal value of \(\Arg(z)\) must lie within \((- \pi, \pi]\).

Answer 1

The Correct Option is C

Step 1: Write z1 and z2 using their given arguments.
Let \(z_1=r_1 e^{i\pi/3}\) and \(z_2=r_2 e^{i\pi/5}\) for some positive moduli \(r_1,r_2\).

Step 2: Combine the arguments using the quotient rule.
\(\operatorname{Arg}\left(\frac{z_1}{z_2}\right)=\operatorname{Arg}z_1-\operatorname{Arg}z_2=\frac{\pi}{3}-\frac{\pi}{5}\)

Step 3: Simplify the fraction.
\[ \frac{\pi}{3}-\frac{\pi}{5}=\frac{5\pi-3\pi}{15}=\frac{2\pi}{15} \]

Step 4: Conclusion.
\[ \boxed{\frac{2\pi}{15}} \]