https://file.aiquickdraw.com/mathbot/file/1728100284922db6aqi8n.jpg

To convert polar equations to rectangular form, we use the following identities that relate polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\):

1. \( x = r \cos \theta \)
2. \( y = r \sin \theta \)
3. \( r^2 = x^2 + y^2 \)

Now, let's solve each part.

### (a) \( r \sin \theta = -3 \)
Using the identity \( y = r \sin \theta \), we can rewrite the equation as:
\[
y = -3
\]
This is the rectangular form.

### (b) \( r = 4 \)
Using the identity \( r^2 = x^2 + y^2 \), square both sides of the equation:
\[
r^2 = 4^2 = 16
\]
So, the rectangular form is:
\[
x^2 + y^2 = 16
\]
This represents a circle with a radius of 4 centered at the origin.

---

Would you like more details or clarifications on these steps? 

Here are five related questions to expand on this topic:
1. How do we convert rectangular coordinates to polar coordinates?
2. What is the general form of a polar equation for a line?
3. How can you graph the equation \( x^2 + y^2 = 16 \)?
4. What are some common polar-to-rectangular conversions for different curves?
5. How does the unit circle help in converting between polar and rectangular forms?

**Tip:** Whenever converting between coordinate systems, it's crucial to remember the core identities that link polar coordinates (\(r, \theta\)) to rectangular coordinates (\(x, y\)). These identities simplify the transformation process.

Convert each polar equation to rectangular form. (a) r * sin(θ) = -3, (b) r = 4

Learn how to convert polar equations r * sin(θ) = -3 and r = 4 into rectangular form. This problem uses key coordinate geometry concepts, including the formulas x = r * cos(θ) and y = r * sin(θ), suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation