x^2+(y-5)^2 = 25
x^2+y^2-10y+25 = 25
x^2+y^2 = 10y
r^2 = 10rsin(theta)
r = 10sin(theta)
Show all your work. Your final answer should be r=
Don't skip any steps and don't explain with words, show the steps of the equation.
x^2+y^2-10y+25 = 25
x^2+y^2 = 10y
r^2 = 10rsin(theta)
r = 10sin(theta)
Step 1: Replace x with r * cos(θ) and y with r * sin(θ).
(r * cos(θ))^2 + (r * sin(θ) - 5)^2 = 25
Step 2: Expand the equation.
r^2 * cos^2(θ) + (r * sin(θ) - 5)^2 = 25
Step 3: Simplify and apply trigonometric identities.
r^2 * cos^2(θ) + r^2 * sin^2(θ) - 10r * sin(θ) + 25 = 25
Step 4: Combine like terms.
r^2 * (cos^2(θ) + sin^2(θ)) - 10r * sin(θ) + 25 = 25
Step 5: Apply the trigonometric identity cos^2(θ) + sin^2(θ) = 1.
r^2 - 10r * sin(θ) + 25 = 25
Step 6: Simplify.
r^2 - 10r * sin(θ) = 0
Finally, rearranging the equation gives us the polar form:
r = 10 * sin(θ)
Hence, the final answer is r = 10 * sin(θ).