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

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## Answer to a math question Plot the polar equation 𝒓 = 𝟑 − 𝟑 𝐬𝐢𝐧$𝜽$. Discuss the Limaçon's loop size, direction, and key points, including the loops

Seamus
4.9
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.

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 $3, 360$.
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