Find a Cartesian equation for
r= 3sin deta
http://www.teacherschoice.com.au/maths_library/coordinates/polar_-_rectangular_conversion.htm
x=rcosTheta
y=rsinTheta
To convert the equation from polar form (using the polar coordinates r and θ) to Cartesian form (using the Cartesian coordinates x and y), we need to use the following relationships:
x = r * cos(θ)
y = r * sin(θ)
For the given polar equation r = 3sin(θ), we can substitute these values into the Cartesian equations:
x = (3sin(θ)) * cos(θ)
y = (3sin(θ)) * sin(θ)
Simplifying these expressions:
x = 3sin(θ)cos(θ)
y = 3sin^2(θ)
Therefore, the Cartesian equation equivalent to r = 3sin(θ) is:
x = 3sin(θ)cos(θ)
y = 3sin^2(θ)
To find a Cartesian equation for a polar equation, we can use the following conversion formulas:
x = r * cos(theta)
y = r * sin(theta)
Given the polar equation r = 3sin(theta), we can substitute these conversion formulas to get the corresponding Cartesian equation.
x = (3sin(theta)) * cos(theta)
y = (3sin(theta)) * sin(theta)
Simplifying these equations, we get:
x = 3sin(theta)cos(theta)
y = 3sin^2(theta)
Now, since we want the Cartesian equation in terms of x and y, we need to eliminate the theta variable. We can do this by using a trigonometric identity:
sin^2(theta) + cos^2(theta) = 1
Rearranging this equation, we get:
sin^2(theta) = 1 - cos^2(theta)
Substituting this into the equation for y, we have:
y = 3(1 - cos^2(theta))
Now, let's substitute for sin(theta) in the equation for x using the same trigonometric identity:
sin(theta) = sqrt(1 - cos^2(theta))
x = 3(sqrt(1 - cos^2(theta))) * cos(theta)
Combining these equations, we have:
x = 3sqrt(1 - cos^2(theta)) * cos(theta)
y = 3(1 - cos^2(theta))
Therefore, the Cartesian equation for the polar equation r = 3sin(theta) is:
x = 3sqrt(1 - (y/3)^2)
y = 3(1 - x^2/9)