To convert a polar equation to a Cartesian equation, we can make use of the relationships between polar and Cartesian coordinates. The conversion can be done using the following trigonometric identities
x = r·cos(θ)
y = r·sin(θ)
Let's apply these conversions to the given polar equations:
a. r = 2:
To convert this equation, we can substitute x = r·cos(θ) and y = r·sin(θ):
x = 2·cos(θ)
y = 2·sin(θ)
Therefore, the Cartesian equation for r = 2 is x^2 + y^2 = 4.
b. r = 3sin(θ):
Substituting x = r·cos(θ) and y = r·sin(θ) in the equation:
x = 3sin(θ)·cos(θ)
y = 3sin^2(θ)
Using trigonometric identities, we can rewrite:
2y = 3sin(2θ)
This is the Cartesian equation for r = 3sin(θ).
c. r^2 = sin(2θ):
Square both sides of the equation:
r^2 = sin(2θ)
(r·cos(θ))^2 + (r·sin(θ))^2 = sin(2θ)
Using trigonometric identities, we can simplify further:
x^2 + y^2 = sin(2θ)
So, the Cartesian equation for r^2 = sin(2θ) is x^2 + y^2 = sin(2θ).
By applying the respective substitutions and simplifications, we have obtained the Cartesian equations for the given polar equations.