What does the polar equation #theta=arcsin(1-3r)# represent?

Question

Ezekiel Alexander · Accepted Answer

undefined The polar equation $ \theta = \arcsin(1 - 3r) $ represents a curve in polar coordinates. This equation describes a spiral that starts at the origin (pole) and spirals outward as $ r $ increases. The angle $ \theta $ depends on the value of $ r $, where $ r $ represents the distance from the origin, and $ \theta $ represents the angle between the initial ray and the radius vector.

Cameron Alexander · Answer

Please see below.

I is not clear what is exactly desired by the questioner. However, #theta=arcsin(1-3r)# means #sintheta=1-3rhArrrsintheta=r-3r^2# or #3r=1-sintheta#, which is a polar equation of a cardioid, a heart shaped figure, whose graph appears as one given below. As polar coordinates #(r,theta)# are related to Cartesian coordinates #(x,y)# by relations #x=rcostheta#, #y=rsintheta# and #r^2=x^2+y^2# #theta=arcsin(1-3r)# is equivalent to #y=sqrt(x^2+y^2)-3(x^2+y^2)# graph{y=sqrt(x^2+y^2)-3(x^2+y^2) [-1.268, 1.232, -0.88, 0.37]}