y = r sinθ
x = r cosθ
r^2 sin^2θ = r cosθ
r = cosθ/sin^2θ = cscθ cotθ
the rectangular equation.
y^2 = x
x = r cosθ
r^2 sin^2θ = r cosθ
r = cosθ/sin^2θ = cscθ cotθ
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!
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.
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.