See
https://socratic.org/questions/how-do-you-create-a-graph-of-r-3-sin-7theta-5
0 <= r <= 50≤r≤5.
The period is (8/3)pi(83)π.. No graph for negative r., when
(3/4)theta in ( pi, 2pi ) rArr theta in ((4/3)pi, (8/3)pi)(34)θ∈(π,2π)⇒θ∈((43)π,(83)π).
For four complete rotations, through 8pi8π, three loop sought to
be created.
Upon setting (8/3)pi = k(2pi)(83)π=k(2π), for least integer k, ,k = 3 #..
Use sin 3theta = sin 4(3/4theta)sin3θ=sin4(34θ) and expand both.
3 cos^2theta sin theta - sin^3theta3cos2θsinθ−sin3θ
= 4 (cos^3(3/4theta) sin(3/4theta)=4(cos3(34θ)sin(34θ)
- cos(3/4theta) sin^3(3/4theta)−cos(34θ)sin3(34θ)
Now switch over to Cartesian form, using astutely
sin(3/4theta) = r / 5, cos(3/4theta) = sqrt ( 1 - r^2/25),sin(34θ)=r5,cos(34θ)=√1−r225,
r (cos theta, sin theta ) = ( x, y ) and r= sqrt ( x^2 + y^3 )r(cosθ,sinθ)=(x,y)andr=√x2+y3, to get
3 x^2 y - y^3 = 4 ( x^2 + y^2 )^1.5(( 1 - (x^2 + y^2 )/25)^1.53x2y−y3=4(x2+y2)1.5((1−x2+y225)1.5
( (x^2 + y ^2)/25 )^0.5 -( 1 - (x^2 + y^2 )/25)^0.5(x2+y225)0.5−(1−x2+y225)0.5
(( x^2 + y ^2 )/25)^1.5)(x2+y225)1.5)
I have paved the way. the Socratic graph is immediate.
graph{3x^2 y-y^3 - 0.0064 ( x^2 + y^2 )^2( 25 - (x^2 + y^2 ))^0.5( 25 - 2(x^2 + y^2 ) ) = 0[-10 10 -5 5]}
See graph of r = cos (3/4)thetar=cos(34)θ:
graph{x^3-3xy^2-(x^2+y^2)^1.5(8(x^2+y^2)^2-8(x^2+y^2)+1)=0[-2 2 -1 1]}
