The letters r and theta represent polar coordinates. Write each equation in rectangular coordinates (x, y) form.
(1) r = 4
(2) r = 3/(3 - cos(t)), where t = theta
To convert the polar equations to rectangular coordinates, we can use the relationships between the polar coordinates (r, θ) and rectangular coordinates (x, y):
1. For the given equation r = 4:
Using the relationships, we have:
x = r * cos(θ)
y = r * sin(θ)
Substituting r = 4 into the equations, we get:
x = 4 * cos(θ)
y = 4 * sin(θ)
Therefore, the equation in rectangular coordinates is: x = 4 * cos(θ), y = 4 * sin(θ)
2. For the equation r = 3 / (3 - cos(θ)), where t = θ:
First, let's simplify the equation:
Multiplying both sides by 3 - cos(θ), we get:
r * (3 - cos(θ)) = 3
3r - r*cos(θ) = 3
Using the relationships, we have:
x = r * cos(θ)
y = r * sin(θ)
Substituting r*cos(θ) with x and r*sin(θ) with y, we get:
3r - x = 3
Rearranging the equation, we have:
x = 3r - 3
Therefore, the equation in rectangular coordinates is: x = 3r - 3.