Convert the following in polar form:
`(1+3i)/(1-2i)`

Answer

To convert the complex number \((1 + 3i)/(1 - 2i)\) into polar form, we will follow these steps:

### Step 1: Simplify the Complex Number
We start with the expression \((1 + 3i)/(1 - 2i)\). To simplify this, we will multiply the numerator and the denominator by the conjugate of the denominator, which is \((1 + 2i)\).

\[ \frac{(1 + 3i)(1 + 2i)}{(1 - 2i)(1 + 2i)} \]

### Step 2: Calculate the Denominator
Now, we calculate the denominator:

\[ (1 - 2i)(1 + 2i) = 1^2 - (2i)^2 = 1 - 4(-1) = 1 + 4 = 5 \]

### Step 3: Calculate the Numerator
Next, we calculate the numerator:

\[ (1 + 3i)(1 + 2i) = 1 \cdot 1 + 1 \cdot 2i + 3i \cdot 1 + 3i \cdot 2i = 1 + 2i + 3i + 6(-1) = 1 + 5i - 6 = -5 + 5i \]

### Step 4: Combine the Results
Now we can combine the results from the numerator and denominator:

\[ \frac{-5 + 5i}{5} = -1 + i \]

### Step 5: Write in Standard Form
Now we have the complex number in standard form:

\[ -1 + i \]

### Step 6: Convert to Polar Form
To convert \(-1 + i\) into polar form, we need to find the modulus \(r\) and the argument \(\theta\).

#### Step 6.1: Calculate the Modulus \(r\)
The modulus \(r\) is given by:

\[ r = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

#### Step 6.2: Calculate the Argument \(\theta\)
The argument \(\theta\) is given by:

\[ \tan(\theta) = \frac{\text{Imaginary part}}{\text{Real part}} = \frac{1}{-1} = -1 \]

This corresponds to an angle in the second quadrant, where \(\tan(\theta) = -1\). The angle that satisfies this is:

\[ \theta = \frac{3\pi}{4} \]

### Step 7: Write the Polar Form
Now we can write the polar form of the complex number:

\[ \sqrt{2} \left(\cos\left(\frac{3\pi}{4}\right) + i\sin\left(\frac{3\pi}{4}\right)\right) \]

### Final Answer
Thus, the polar form of \(\frac{(1 + 3i)}{(1 - 2i)}\) is:

\[ \sqrt{2} \left(\cos\left(\frac{3\pi}{4}\right) + i\sin\left(\frac{3\pi}{4}\right)\right) \]

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