Studdy AI Logo
Math  /  Algebra

QuestionFind a polar equation for the curve represented by the given Cartesian equation. (Assume 0≤θ<2π0 \leq \theta<2 \pi.) x2+y2=6yx^{2}+y^{2}=6 y

Studdy Solution

STEP 1

1. The given Cartesian equation is x2+y2=6y x^2 + y^2 = 6y .
2. We need to convert this equation into polar coordinates.
3. The polar coordinates are related to Cartesian coordinates by the equations x=rcos⁡θ x = r \cos \theta and y=rsin⁡θ y = r \sin \theta .
4. The equation x2+y2=r2 x^2 + y^2 = r^2 holds in polar coordinates.

STEP 2

1. Express the Cartesian equation in terms of polar coordinates.
2. Simplify the polar equation.

STEP 3

Start by substituting the polar coordinate expressions for x x and y y into the Cartesian equation:
x2+y2=6y x^2 + y^2 = 6y
Substitute x=rcos⁡θ x = r \cos \theta and y=rsin⁡θ y = r \sin \theta :
(rcos⁡θ)2+(rsin⁡θ)2=6(rsin⁡θ) (r \cos \theta)^2 + (r \sin \theta)^2 = 6(r \sin \theta)

STEP 4

Simplify the equation using the identity x2+y2=r2 x^2 + y^2 = r^2 :
r2=6rsin⁡θ r^2 = 6r \sin \theta

STEP 5

To solve for r r , divide both sides by r r , assuming r≠0 r \neq 0 :
r=6sin⁡θ r = 6 \sin \theta
The polar equation for the curve is:
r=6sin⁡θ r = 6 \sin \theta

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord