Final answer:
To convert the polar equation r = 3 sin(θ) to a cartesian equation, substitute the polar coordinate representations into the sine and cosine-related expressions and simplify to obtain the formula x^2 + y^2 = 9, indicating a circle with radius 3.
Step-by-step explanation:
To rewrite the polar equation r = 3 sin(θ) as a cartesian equation, we use the relationships x = r cos(θ) and y = r sin(θ). Plugging the given polar equation into these relationships, we replace r sin(θ) with y and get y = 3 sin(θ). This is equal to our original polar expression, meaning we can substitute r with its cartesian equivalent, \( \sqrt{x^2 + y^2} \).
But since r = y, we can square both sides to get y^2 = 9 sin^2(θ). Using our second relationship, sin(θ) = y/r, we substitute r with \( \sqrt{x^2 + y^2} \) to give y^2 = 9 (y/&sqrt{x^2 + y^2})^2. Simplifying this, we get y^2 = 9y^2/(x^2 + y^2), and after multiplying both sides by x^2 + y^2, we have y^2(x^2 + y^2) = 9y^2.
After cancelling out y^2 from both sides, we end up with the cartesian equation x^2 + y^2 = 9, which represents a circle with a radius of 3 centered at the origin. Note that because we started with a sine function that dealt with the y-coordinate in the polar system, it makes sense that our final equivalent equation is a standard circle equation in cartesian coordinates centered on the y-axis.