Question

How do you graph `r=4+7sintheta`?

Answer 1

See graph and details.

Explanation:

Using `( x, y ) = r ( cos theta, sin theta )`, the Cartesian form of

`r = 4 + 7 sin theta` is got as

`x^2 + y^2 = 4 sqrt ( x^2 + y^2 ) +7 y`.

Here, `r = x^2 + y^2 >= 0` and `r in [0, 11]`.

As `cos (-theta) = cos theta`, the graph is symmetrical about the

initial line `theta = 0`.

Graph of the limacon `r = 4 + 7 sin theta`:
graph{ x^2 + y^2 - 4 sqrt ( x^2 + y^2 ) -7 y = 0[-14 14 -2 12]}

This limacon, with the characteristic dimple, is the member n = 1 of

the multiloop family

`r = 4 + 7 sin n theta, n = 1, 2, 3, ...`. See dimple-free graphs for n

= 2 and 7.
graph{ (x^2+y^2)^1.5- 4(x^2+y^2)-7xy=0[-24 24 -12 12]}
graph{ (x^2+y^2)^4 - 4(x^2+y^2)^3.5 -7(x^7-21x^5y^2+35x^3y^4-7xy^6)=0[-24 24 -12 12]}