Find the modulus and amplitude of `(2 + 3i)/(3+2i)`

Answer

To find the modulus and amplitude of the complex number \(\frac{2 + 3i}{3 + 2i}\), we will follow these steps:

### Step 1: Find the Modulus
The modulus of a complex number \( z = x + yi \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]
For the complex numbers \( z_1 = 2 + 3i \) and \( z_2 = 3 + 2i \), we can find their moduli separately.

**Modulus of \( z_1 \):**
\[ |z_1| = |2 + 3i| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \]

**Modulus of \( z_2 \):**
\[ |z_2| = |3 + 2i| = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \]

Now, using the property of moduli:
\[ \left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|} = \frac{\sqrt{13}}{\sqrt{13}} = 1 \]

### Step 2: Find the Amplitude
The amplitude (or argument) of a complex number \( z = x + yi \) is given by:
\[ \text{arg}(z) = \tan^{-1}\left(\frac{y}{x}\right) \]

**Amplitude of \( z_1 \):**
\[ \text{arg}(z_1) = \tan^{-1}\left(\frac{3}{2}\right) \]

**Amplitude of \( z_2 \):**
\[ \text{arg}(z_2) = \tan^{-1}\left(\frac{2}{3}\right) \]

Using the property of amplitudes:
\[ \text{arg}\left(\frac{z_1}{z_2}\right) = \text{arg}(z_1) - \text{arg}(z_2) \]
Thus,
\[ \text{arg}\left(\frac{2 + 3i}{3 + 2i}\right) = \tan^{-1}\left(\frac{3}{2}\right) - \tan^{-1}\left(\frac{2}{3}\right) \]

Using the formula for the difference of two arctangents:
\[ \tan^{-1}(x) - \tan^{-1}(y) = \tan^{-1}\left(\frac{x - y}{1 + xy}\right) \]
where \( x = \frac{3}{2} \) and \( y = \frac{2}{3} \):
\[ \text{arg}\left(\frac{2 + 3i}{3 + 2i}\right) = \tan^{-1}\left(\frac{\frac{3}{2} - \frac{2}{3}}{1 + \left(\frac{3}{2}\right)\left(\frac{2}{3}\right)}\right) \]

Calculating the numerator:
\[ \frac{3}{2} - \frac{2}{3} = \frac{9 - 4}{6} = \frac{5}{6} \]

Calculating the denominator:
\[ 1 + \left(\frac{3}{2}\right)\left(\frac{2}{3}\right) = 1 + 1 = 2 \]

Thus,
\[ \text{arg}\left(\frac{2 + 3i}{3 + 2i}\right) = \tan^{-1}\left(\frac{\frac{5}{6}}{2}\right) = \tan^{-1}\left(\frac{5}{12}\right) \]

### Final Result
The modulus and amplitude of \(\frac{2 + 3i}{3 + 2i}\) are:
- Modulus: \( 1 \)
- Amplitude: \( \tan^{-1}\left(\frac{5}{12}\right) \)