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Convert the polar equation r = 16 sin θ to a rectangular equation.

Convert the polar equation r = 16 sin θ to a rectangular equation.-example-1
User Andrei Vinogradov
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2 Answers

11 votes
11 votes

After converting polar equation r =
16 sin \theta to a rectangular equation we get
x^2 + (y - 8)^2 = 16. Option C is the right choice.

In rectangular coordinates, x = r cos(θ) and y = r sin(θ). Substituting these into the polar equation r = 16 sin(θ), we get:

x = 16 sin(θ) cos(θ)

y =
16 sin^2(\theta)

Using the double-angle identity

sin(2θ) = 2 sin(θ) cos(θ), we can rewrite x as:

x = 8 sin(2θ)

Squaring both equations and adding them, we get:


x^2 + (y - 8)^2 = 16^2

Therefore, the answer is:
x^2 + (y - 8)^2 = 16

Option C is the right choice.

User Tempid
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2.9k points
7 votes
7 votes

From the conversion formula, given by


y=rsin\theta

we have that


sin\theta=(y)/(r)

By substituting this result into the given polar equation, we have


r=16*(y)/(r)

So, by multiplying both sides by r, we get


r^2=16y

Now, from the conversion formula


r^2=x^2+y^2

we have that


x^2+y^2=16y

By subtracting 16y to both sides, we obtain


x^2+y^2-16y=0

By completing the square of the quadratic equation on y, we get


x^2+(y-8)^2-64=0

then, by adding 64 to both sides, we get


x^2+(y-8)^2=64

Therefore, the answer is the last option.

User Emersonthis
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3.4k points