Question

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

71

likes
357 views

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

Expert avatar
Seamus
4.9
26 Answers
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).

Frequently asked questions (FAQs)
Math question: What is the value of log base 5 of (25^2) + log base 2 of 16 - log base 7 of (49/7)?
+
What is the value of √169?
+
What is the limit as x approaches 5 of (2x+1)/(x-5)?
+
New questions in Mathematics
I) Find the directional derivative of 𝑓(𝑥, 𝑦) = 𝑥 sin 𝑦 at (1,0) in the direction of the unit vector that make an angle of 𝜋/4 with positive 𝑥-axis.
If f(x) = 3x 2, what is the value of x so that f(x) = 11?
7273736363-8
Determine the momentum of a 20 kg body traveling at 20 m/s.
Perpetual annuities are a series of payments whose duration has no end. Explain how can we calculate them, if they have no end?
2x+4x=
reduce the expression (7.5x 12)÷0.3
The ninth term of a given geometric progression, with reason q , is 1792, and its fourth term is 56. Thus, calculate the fourth term of another geometric progression, whose ratio is q +1 and whose first term is equal to the first term of the first P.G. described.
89, ÷ 10
3.24 ÷ 82
Let v be the set of all ordered pairs of real numbers and consider the scalar addition and multiplication operations defined by: u+v=(x,y)+(s,t)=(x+s+1,y+t -two) au=a.(x,y)=(ax+a-1,ay-2a+2) It is known that this set with the operations defined above is a vector space. A) calculate u+v is au for u=(-2,3),v=(1,-2) and a=2 B) show that (0,0) #0 Suggestion find a vector W such that u+w=u C) who is the vector -u D) show that axiom A4 holds:-u+u=0
2x2
Determine the Linear function whose graph passes through the points (6, -2) and has slope 3.
Kaya deposits 25,000 into an account that earns 3% interest compounded monthly. How much does Kaya have in the account after 6 years 8 months? Round to the nearest cent. 32,912.50 30,000 29,923.71 30,527.45
In poker, a full house consists of five cards, where two of the cards have the same number (or letter) and the remaining three also have the same number (or letter) as each other (but not as the previous two cards). Use a search engine or Wikipedia to understand the concept better if necessary. In how many different ways can one obtain a full house?
-5x=115
2x-4=8
How much does 7.2 moles of ammonium dichromate weigh? (NH4)2Cr2O7
Paul invites 12 friends to his birthday. He wants to give 15 candies to everyone two. The candies are sold in packs of 25. How many should he buy? packages?
To apply a diagnostic test, in how many ways can 14 students be chosen out of 25? if the order does not matter