Find an equation in cylindrical coordinates for
the rectangular equation.
y^2 = x
y = r sinθ
x = r cosθ
r^2 sin^2θ = r cosθ
r = cosθ/sin^2θ = cscθ cotθ
Why did the mathematician throw a drowning polynomial into the ocean? Because it didn't have a real solution! Anyways, let me help you out with your question.
In cylindrical coordinates, we have:
x = r*cos(theta)
y = r*sin(theta)
So, to convert the rectangular equation y^2 = x into cylindrical coordinates, we substitute x and y:
(y*sin(theta))^2 = r*cos(theta)
Simplifying, we get:
r^2*sin^2(theta) = r*cos(theta)
And that's the equation in cylindrical coordinates for the given rectangular equation. Just a heads up though, it might not be as straightforward as a clown hopping on a unicycle, so be cautious when applying it!
To find the cylindrical equation for the rectangular equation, we need to convert the variables from Cartesian coordinates (x, y) to cylindrical coordinates (ρ, θ, z).
In cylindrical coordinates, ρ represents the distance from the origin to a point in the x-y plane, θ represents the angle the line connecting the origin and the point makes with the positive x-axis, and z represents the same as in Cartesian coordinates.
We can rewrite the rectangular equation y^2 = x as:
ρsin(θ)^2 = ρcos(θ)
To simplify this equation, we can divide both sides by ρ:
sin(θ)^2 = cos(θ)
Then, we can manipulate the equation to get it in a more standard form:
sin(θ)^2 - cos(θ) = 0
sin(θ)^2 - 1 + sin^2(θ) = 0
2sin(θ)^2 - 1 = 0
Finally, we have obtained the equation in cylindrical coordinates:
2ρsin(θ)^2 - ρ = 0
Therefore, the equation in cylindrical coordinates for y^2 = x is 2ρsin(θ)^2 - ρ = 0.
To find the equation in cylindrical coordinates corresponding to the given rectangular equation, we need to express the variables (x, y) in terms of the cylindrical coordinates (ρ, θ, z).
In cylindrical coordinates:
- ρ represents the distance from the origin to the point projected onto the xy-plane (similar to the distance from the origin to a point in polar coordinates).
- θ represents the angle measured from the positive x-axis to the line segment connecting the origin to the point.
- z represents the height or displacement from the xy-plane.
To express (x, y) in terms of (ρ, θ, z), we can use the following relationships:
- x = ρ * cos(θ)
- y = ρ * sin(θ)
Let's substitute these expressions into the given equation y^2 = x:
(ρ * sin(θ))^2 = ρ * cos(θ)
Simplifying the equation:
ρ^2 * sin^2(θ) = ρ * cos(θ)
Now, let's convert this equation into a more standard form by dividing both sides by ρ:
ρ * sin^2(θ) = cos(θ)
This is the equation in cylindrical coordinates corresponding to the given rectangular equation y^2 = x.