Question

How do you graph `r=2sin2x`?

Answer 1

See graph and explanation.

Explanation:

The graph of `r = a sin ( ntheta - alpha )`, n = 1, 2, 3, 4, ...# shows

n equal and symmetrical loops, around the pole r = 0.

The graph of `r = a sin ( ntheta - alpha )` is obtained by rotating

anticlockwise graph of `r = a sin theta`, about `theta = 0`,

through `alpha`.

Use `( x, y ) = r ( cos theta, sin theta ), r = sqrt( x^2 + y^2 )` and

`sin 2theta = 2 sin theta cos theta` and get the Cartesian form of

`r = 2 sin 2theta` as

`( x^2 + y^2 )^1.5 - 4 xy = 0`.

Now, the Socratic graph is immediate.
graph{ (x^2 + y^2 )^1.5 - 4 xy = 0[-4 4 -2 2 ]}

For anticlockwise rotation, through `alpha = p/2`, use

`r = 2 sin ( 2 ( theta - pi/2)) = - 2 sin 2theta`

The graph is immediate.
graph{ (x^2 + y^2 )^1.5 + 4 xy = 0[-4 4 -2 2 ]}

For clockwise rotation, through `alpha = pi/4`, use

`r = 2 sin ( 2 ( theta + pi/4)) = 2 cos 2theta`.

See the graph, using

`cos 2theta = (cos^2theta - sin^2theta) = ( x^2 - y^2 )/r^2`

graph{(x^2+y^2)^1.5-2(x^2-y^2)=0[-4 4 -2 2]}