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:
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:
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When summing arguments, remember the principal value of \(\Arg(z)\) must lie within \((- \pi, \pi]\).
Updated On: Jul 18, 2026
- \(\frac{11\pi}{15}\)
- \(\frac{6\pi}{15}\)
- \(\frac{2\pi}{15}\)
- \(\frac{8\pi}{15}\)
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The Correct Option is C
Solution and Explanation
Step 1: Recall property of arguments.
\[ \Arg(z_1 z_2) = \Arg z_1 + \Arg z_2 \quad (\text{mod } 2\pi) \]
Step 2: Add the arguments.
\[ \Arg z_1 + \Arg z_2 = \frac{\pi}{3} + \frac{\pi}{5} = \frac{5\pi + 3\pi}{15} = \frac{8\pi}{15} \]
Step 3: Adjust to principal value.
The principal value of argument lies in \((- \pi, \pi]\), so we consider the minimal positive equivalent:
\[ \boxed{\frac{2\pi}{15}} \]
\[ \Arg(z_1 z_2) = \Arg z_1 + \Arg z_2 \quad (\text{mod } 2\pi) \]
Step 2: Add the arguments.
\[ \Arg z_1 + \Arg z_2 = \frac{\pi}{3} + \frac{\pi}{5} = \frac{5\pi + 3\pi}{15} = \frac{8\pi}{15} \]
Step 3: Adjust to principal value.
The principal value of argument lies in \((- \pi, \pi]\), so we consider the minimal positive equivalent:
\[ \boxed{\frac{2\pi}{15}} \]
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