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

To find the argument of the complex number \( \frac{3}{5}+\frac{4 i}{5} \), we can express it in the form \( a + bi \), where \( a = \frac{3}{5} \) and \( b = \frac{4}{5} \). The argument \( \theta \) can be found using the formula \( \theta = \tan^{-1}\left(\frac{b}{a}\right) = \tan^{-1}\left(\frac{\frac{4}{5}}{\frac{3}{5}}\right) = \tan^{-1}\left(\frac{4}{3}\right) \). To solve for the argument, we know that \( \tan^{-1}\left(\frac{4}{3}\right) \) relates to the angles in a right triangle. If we use the fact that \( \tan(\theta) = \frac{opposite}{adjacent} \), we can conclude that this angle is situated in the first quadrant, making direct use of the inverse tangent function applicable. Further simplification isn't necessary, but knowing the right triangles and the commonly known angles can help! Ultimately, you should check all given options for equivalence to \( \tan^{-1}\left(\frac{4}{3}\right) \) in terms of arctangent values. The result, upon checking various options closely indicates: \(\text{Select option b: } 2 \arctan\left(\frac{1}{3}\right).\)