To convert each of the polar equations to rectangular form, we will use the following formulas:
r cos θ = x
r sin θ = y
r cos θ = 3
Using the above formulas, we can rewrite this equation in rectangular form:
r cos θ = x
r = √(x^2 + y^2) (from the Pythagorean theorem)
cos θ = x/r
Substituting these values, we get:
x = r cos θ = 3
y = r sin θ = √(r^2 - x^2) = √(9 - 9) = 0
Therefore, the rectangular form of the polar equation r cos θ = 3 is:
x = 3, y = 0
r = 2 sin θ
Again, using the formulas mentioned above, we can rewrite this equation in rectangular form:
r sin θ = y
r = √(x^2 + y^2)
sin θ = y/r
Substituting these values, we get:
x = √(r^2 - y^2) = √(4 sin^2 θ) = 2 sin θ or -2 sin θ
y = r sin θ = 2 sin^2 θ
Therefore, the rectangular form of the polar equation r = 2 sin θ is:
x = 2 sin θ or -2 sin θ, y = 2 sin^2 θ