https://file.aiquickdraw.com/mathbot/file/1728100475416khd4dhaz.jpg

The task in the image is to convert polar equations to rectangular form. To do this, we use the following standard relationships between polar and rectangular coordinates:

\[
x = r \cos \theta, \quad y = r \sin \theta, \quad r^2 = x^2 + y^2
\]

Let's address each part:

### Part (a) \( r \cos \theta = 2 \)

Using the conversion formula \( x = r \cos \theta \), we substitute this into the equation:

\[
x = 2
\]

This is the rectangular form of the equation.

### Part (b) \( r = -6 \)

We use the relationship \( r^2 = x^2 + y^2 \) to convert this into rectangular form. First, square both sides of the equation:

\[
r^2 = (-6)^2 = 36
\]

Now substitute \( r^2 = x^2 + y^2 \):

\[
x^2 + y^2 = 36
\]

This is the rectangular form of the equation.

Let me know if you need further clarification or details.

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Here are 5 related questions you might find interesting:
1. What happens to the rectangular form if we change the angle \(\theta\) in polar coordinates?
2. How would the equation look if the polar form had \(r \sin \theta\)?
3. Can we convert rectangular coordinates back to polar coordinates easily?
4. What is the significance of negative \(r\) in polar coordinates?
5. How would we plot these equations on a graph?

### Tip:
Always remember that \(\theta\) in polar coordinates represents the angle with the positive \(x\)-axis, and adjusting it modifies the direction of the vector from the origin!

Convert the following polar equations to rectangular form: (a) r * cos(θ) = 2 and (b) r = -6

Learn how to convert polar equations to rectangular form using key formulas such as x = r * cos(θ) and r^2 = x^2 + y^2. This problem demonstrates the steps to convert equations like r * cos(θ) = 2 and r = -6. Suitable for students in Grades 10-12.

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