Question

find the points on the curve r=-4sin(theta) where the tangent line is horizontal

Answer 1

i don't know about polar coordinates but if you change r = sqrt(x2+y2) and sin theta to y/(x2+y2) ... then surely it will be easy to do.

Answer 2

well we first convert this polar equation to an equation with only x and y's. now it's probably helpful to sketch it first,

as you can see, our center is at (0,-2) and the radius is 2 so we'll have
x^2+(y+2)^2=4
even without calculus we could still come up with the point where the tangent line is horizontal by analyzing again the graph:

so you can see that it will have tangent line at points (0,0) and (0,-4). And we can confirm this using calculus. now we know that we'll have a tangent line at points where our derivative dy/dx=0. so in this case our dy/dx is:
\[dy/dx=-2x/2(y+1)\]
now if we plug in (0,-4) and (0,0) we'll get dy/dx=0 so these points are really the points where the tangent line will be horizontal. So the answer is:
(0,0) and (0,-4)<-----answer