Plot the polar equation 𝒓 = 𝟑 − 𝟑 𝐬𝐢𝐧(𝜽). Discuss the Limaçon's loop size, direction, and key points, including the loops

To plot the polar equation 𝒓 = 𝟑 − 𝟑 𝐬𝐢𝐧(𝜽), we can break it down into smaller steps:

Step 1: Determine the loop size and direction:
To find the loop size, we can look at the coefficient of the sine function. In this case, the coefficient is -3. This negative coefficient means that the loop will be on the inside of the circle with a radius of 3.

Step 2: Find the key points on the loop:
To find the key points on the loop, we can substitute different values of θ into the equation and evaluate r.

When θ = 0 degrees:
r = 3 - 3 sin(0) = 3 - 3(0) = 3
So the point (3, 0) is on the loop.

When θ = 90 degrees:
r = 3 - 3 sin(90) = 3 - 3(1) = 0
So the point (0, 90) is on the loop.

When θ = 180 degrees:
r = 3 - 3 sin(180) = 3 - 3(0) = 3
So the point (-3, 180) is on the loop.

When θ = 270 degrees:
r = 3 - 3 sin(270) = 3 - 3(-1) = 6
So the point (6, 270) is on the loop.

When θ = 360 degrees:
r = 3 - 3 sin(360) = 3 - 3(0) = 3
So the point (3, 360) is on the loop.

Step 3: Plot the points and connect them:
Using the key points we found, we can now plot them on a polar coordinate system and connect them to form the loop. The loop will be centered at the origin and have a radius of 3.

Step 4: Answer

The Limaçon's loop of the polar equation r = 3 - 3 sin(θ) has a loop size of 3 and is located on the inside of a circle with a radius of 3. The loop passes through the points (3, 0), (0, 90), (-3, 180), (6, 270), and (3, 360).