Question

Let \(z_1 = 3 + 4i\) and \(z_2 = 1 - 2i\). Then the argument of \(\frac{z_1}{z_2}\) is:

• A. \(\tan\left(\frac{10}{7}\right)\)
• B. \(\tan\left(\frac{2}{11}\right)\)
• C. \(\tan\left(\frac{4}{3}\right) + \tan(2)\)
• D. \(\tan\left(\frac{10}{7}\right) + \pi\)

Hint:

Use the property \(\arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2)\) and the \(\tan\) addition formula carefully to find arguments of complex number quotients.

Answer 1

The Correct Option is A

\[ \arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2) \] Calculate arguments: \[ \arg(z_1) = \tan\left(\frac{4}{3}\right), \quad \arg(z_2) = \tan\left(\frac{-2}{1}\right) = -\tan(2) \] Therefore, \[ \arg\left(\frac{z_1}{z_2}\right) = \tan\left(\frac{4}{3}\right) - (-\tan(2)) = \tan\left(\frac{4}{3}\right) + \tan(2) \] Using addition formula: \[ \tan x + \tan y = \tan\left(\frac{x + y}{1 - xy}\right) \quad \text{(adjust for quadrant if necessary)} \] Substitute: \[ x = \frac{4}{3}, \quad y = 2 \] Calculate numerator and denominator: \[ x + y = \frac{4}{3} + 2 = \frac{10}{3} \] \[ 1 - xy = 1 - \frac{4}{3} \times 2 = 1 - \frac{8}{3} = -\frac{5}{3} \] So, \[ \arg\left(\frac{z_1}{z_2}\right) = \tan\left(\frac{\frac{10}{3}}{-\frac{5}{3}}\right) = \tan(-2) \] Since denominator is negative, add \(\pi\): \[ \arg = \tan(2) + \pi = \tan\left(\frac{10}{7}\right) \] Hence option (A) is correct.