Convert r=7cos(theta) into a rectangular equation

In summary, we learned how to convert a polar equation into a rectangular equation, specifically for the equation r=6cos(theta). By using the relationships between x, y, r, and theta, we were able to manipulate the equation to get it into standard form for a circle. The final equation is (x-3)^2+y^2=9, representing a circle with radius 3 units centered at (3,0).
  • #1
Elissa89
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So we're learning to plot polar equations, which easy enough. But I got a question in the homework that wasn't covered in class:

Convert r=7cos(theta) into a rectangular equation. Use x and y values. I know how to convert when it's x=r*cos(theta) or y=r*sin(theta) and r and theta is given. But this is different and I don't know how to do it.
 
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  • #2
Okay, we are given the polar equation:

\(\displaystyle r=6\cos(\theta)\)

Now, from:

\(\displaystyle x=r\cos(\theta)\implies \cos(\theta)=\frac{x}{r}\)

We may write:

\(\displaystyle r=6\left(\frac{x}{r}\right)\)

Multiply through by \(r\):

\(\displaystyle r^2=6x\)

We know:

\(\displaystyle r^2=x^2+y^2\)

Hence, we have:

\(\displaystyle x^2+y^2=6x\)

This would technically suffice, but I would prefer to continue and put into standard form:

\(\displaystyle x^2-6x+y^2=0\)

Complete the square on \(x\):

\(\displaystyle (x-6x+9)+y^2=9\)

\(\displaystyle (x-3)^2+y^2=3^2\)

Now it's easy to see we have a circle of radius 3 units centered at (3,0).
 

Related to Convert r=7cos(theta) into a rectangular equation

1. What is the purpose of converting r=7cos(theta) into a rectangular equation?

The purpose of converting polar equations into rectangular equations is to represent a curve or shape in a different coordinate system. It allows for easier analysis and comparison of different equations.

2. What is the process for converting r=7cos(theta) into a rectangular equation?

The process involves using the following equations to convert from polar coordinates (r, theta) to rectangular coordinates (x, y):
x = r * cos(theta)
y = r * sin(theta)
In this case, since r=7cos(theta), we can substitute it into the equations to get:
x = 7cos(theta) * cos(theta) = 7cos^2(theta)
y = 7cos(theta) * sin(theta) = 7cos(theta)sin(theta)

3. Can r=7cos(theta) be converted into multiple rectangular equations?

Yes, it is possible to convert a polar equation into multiple rectangular equations. This is because a single polar equation can represent multiple curves or shapes in rectangular coordinates.

4. How can I graph r=7cos(theta) in rectangular coordinates?

To graph r=7cos(theta) in rectangular coordinates, you can plot points by substituting different values of theta into the equation and calculating the corresponding values of x and y. Alternatively, you can use a graphing calculator or software to plot the curve.

5. Are there any limitations to converting polar equations into rectangular equations?

Yes, there are some limitations to converting polar equations into rectangular equations. The conversion only works for equations that can be written in the form of r = f(theta), where f(theta) is a function of theta. It may also be difficult to convert equations that involve complex or non-linear functions.

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