Find a polar equation for the curve represented by the given Cartesian equation.
a. y=5
b.2xy=1
I know i posted a question just for the opposite..Im confused on how to convert the other way around as well. any advice?
a) r sin theta = 5
b) 2 r^2 sin theta cos theta
= r^2 sin (2theta) = 1
To convert a Cartesian equation to a polar equation, you can use the following conversions:
1. For a point (x, y) in the Cartesian plane, the corresponding polar coordinates are given by:
- r = √(x^2 + y^2) (distance from the origin)
- θ = arctan(y/x) (angle from the positive x-axis in the counterclockwise direction)
Now, let's convert the given Cartesian equations to polar equations:
a. y = 5:
This equation represents a horizontal line at y = 5, parallel to the x-axis. In polar coordinates, this corresponds to a circle centered at the origin with radius 5.
Therefore, the polar equation for this Cartesian equation is:
r = 5
b. 2xy = 1:
To convert this equation, we can rewrite it using the polar coordinates:
2r*cos(θ)*r*sin(θ) = 1
2r^2*cos(θ)*sin(θ) = 1
Since cos(θ)*sin(θ) = sin(2θ)/2, we can simplify the equation to:
r^2*sin(2θ) = 1/2
This is the polar equation for the given Cartesian equation.
To convert from polar to Cartesian, you can use the following conversions:
- x = r*cos(θ)
- y = r*sin(θ)
If you have a specific polar equation that you would like to convert, feel free to provide the equation and I can help you convert it to Cartesian form.
To convert the Cartesian equation to a polar equation, we can make use of the relationships between Cartesian and polar coordinates.
a. For the equation y = 5:
To convert it to a polar equation, we need to express x and y in terms of r and θ.
Since y is a constant value of 5, it does not vary with θ.
So, we can represent y as y = r sin(θ).
b. For the equation 2xy = 1:
To convert it to a polar equation, we use the conversion formulas.
Substituting x = r cos(θ) and y = r sin(θ) into the equation, we get:
2(r cos(θ))(r sin(θ)) = 1
2r² cos(θ) sin(θ) = 1
Using the trigonometric identity sin(2θ) = 2sin(θ)cos(θ), we can simplify the equation to:
r² sin(2θ) = 1/2
Thus, the polar equations for the given Cartesian equations are:
a. r sin(θ) = 5
b. r² sin(2θ) = 1/2
I hope this clears up any confusion and helps you understand how to convert equations between Cartesian and polar forms.