Transform the equation y = x tan `alpha` to polar form.

Answer

To transform the equation \( y = x \tan \alpha \) into polar form, we can follow these steps:

### Step 1: Substitute Cartesian Coordinates with Polar Coordinates
In polar coordinates, we have the following relationships:
- \( x = r \cos \theta \)
- \( y = r \sin \theta \)

### Step 2: Substitute \( y \) and \( x \) in the Given Equation
We can substitute \( y \) and \( x \) in the original equation:
\[ r \sin \theta = (r \cos \theta) \tan \alpha \]

### Step 3: Simplify the Equation
Now, we can simplify the equation:
\[ r \sin \theta = r \cos \theta \tan \alpha \]
Since \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \), we can rewrite the equation:
\[ r \sin \theta = r \sin \alpha \]
Now, we can divide both sides by \( r \) (assuming \( r \neq 0 \)):
\[ \sin \theta = \sin \alpha \]

### Step 4: Solve for \( \theta \)
From the equation \( \sin \theta = \sin \alpha \), we can conclude that:
\[ \theta = \alpha + n\pi \quad (n \in \mathbb{Z}) \]
This represents the polar form of the equation.

### Final Answer
Thus, the polar form of the equation \( y = x \tan \alpha \) is:
\[ \theta = \alpha + n\pi \quad (n \in \mathbb{Z}) \]

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