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Find the argument of \( \frac{3}{5}+\frac{4 i}{5} \)

Ask by Schofield Lynch. in Ghana
Mar 15,2025

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Answer

The argument of \( \frac{3}{5} + \frac{4i}{5} \) is \( \arctan\left(\frac{4}{3}\right) \).

Solution

**Step 1. Identify the real and imaginary parts** We are given \[ z = \frac{3}{5} + \frac{4i}{5}, \] so the real part is \( x = \frac{3}{5} \) and the imaginary part is \( y = \frac{4}{5} \). **Step 2. Use the definition of the argument** The argument of a complex number \( z = x + yi \) is defined as \[ \theta = \arctan\left(\frac{y}{x}\right), \] provided that \( x > 0 \) (which is true here). **Step 3. Substitute the values** Substitute \( x = \frac{3}{5} \) and \( y = \frac{4}{5} \) into the formula: \[ \theta = \arctan\left(\frac{\frac{4}{5}}{\frac{3}{5}}\right) = \arctan\left(\frac{4}{3}\right). \] **Final Answer** \[ \theta = \arctan\left(\frac{4}{3}\right) \]

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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, which is given by \( \tan^{-1} \left(\frac{b}{a}\right) \), where \( a \) is the real part and \( b \) is the imaginary part. Here, \( a = \frac{3}{5} \) and \( b = \frac{4}{5} \). First, we compute the ratio: \[ \frac{b}{a} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \] Now, we find the angle whose tangent is \( \frac{4}{3} \): \[ \theta = \tan^{-1} \left(\frac{4}{3}\right) \] The argument of the complex number is therefore: \[ \text{arg}(z) = \tan^{-1} \left(\frac{4}{3}\right) \] This angle is in the first quadrant since both the real and imaginary parts are positive. Hence, the argument is simply: \[ \text{arg}(z) \approx 0.93 \text{ radians} \text{ or } 53.13^\circ \]

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