Find the argument of \( \frac{3}{5}+\frac{4 i}{5} \) Select one: a. \( \quad 2 \arctan \left(\frac{1}{2}\right) \) b. \( \quad \arctan \left(\frac{1}{3}\right) \) c. \( \quad 2 \arctan \left(\frac{1}{3}\right) \) d. \( \quad \arctan \left(\frac{1}{2}\right) \)

To find the argument of the complex number \( z = \frac{3}{5} + \frac{4i}{5} \), we can use the formula for the argument of a complex number, given by \( \text{arg}(z) = \arctan\left(\frac{y}{x}\right) \), where \( x \) is the real part and \( y \) is the imaginary part. Here, \( x = \frac{3}{5} \) and \( y = \frac{4}{5} \). Therefore, \[ \text{arg}(z) = \arctan\left(\frac{\frac{4}{5}}{\frac{3}{5}}\right) = \arctan\left(\frac{4}{3}\right). \] Now, we can convert \( \arctan\left(\frac{4}{3}\right) \) into terms of \( \arctan\left(\frac{1}{3}\right) \) and \( \arctan\left(\frac{1}{2}\right) \): Using the tangent addition formula, we observe that \( \frac{4}{3} = \frac{2}{1} \), which suggests that \( \text{arg}(z) = \arctan(2) \). To connect back with the options provided: \[ \arctan(2) = 2 \arctan\left(\frac{1}{2}\right), \] and thus: The argument can be simplified to \( 2 \arctan\left(\frac{1}{2}\right) \), making the answer option **a**: \( \quad 2 \arctan \left(\frac{1}{2}\right) \) is correct.