Tom K. answered • 07/25/20
Knowledgeable and Friendly Math and Statistics Tutor
r = sin(3θ)
dr = 3cos(3θ)dθ
x = r cos(θ)
y = r sin(θ)
dx = cos(θ) dr - r sin(θ) dθ
y = r sin(θ)
dy = sin(θ) dr + r cos(θ) dθ
Thus, dy/dx = (sin(θ) dr + r cos(θ) dθ)/(cos(θ) dr - r sin(θ) dθ) = , substituting dr = 3 cos(θ)
(3sin(θ) cos(3θ)dθ + r cos(θ) dθ)/(3cos(θ)cos(3θ) dθ - r sin(θ) dθ) = ,substituting r = sin(3θ),
(3sin(θ) cos(3θ)dθ + sin(3θ) cos(θ) dθ)/(3cos(θ)cos(3θ) dθ - sin(3θ) sin(θ) dθ) = , as dθ is in all terms,
(3sin(θ) cos(3θ) + sin(3θ) cos(θ) )/(3cos(θ)cos(3θ) - sin(3θ) sin(θ) )
At θ=π/4, this equals (3* √2/2 * (-√2/2) + √2/2 * √2/2 )/(3*√2/2*(-√2/2) - √2/2 * √2/2 ) =
(-3/2 + 1/2)/(-3/2 - 1/2) = -1/-2 = 1/2
Henry G.
Thank you for your help Tom. My teacher didn't teach me how to do this in class, but rather on my own. I do get a little confused as to why after you found dr you got an x angle and a y angle and got the dx after that step. And also how you got y=r sin(θ) after that step as well. Thanks Tom I learned a lot from this problem!07/26/20