Polar coordinates.?

Question

Im in calculus and we are doing tangent lines to polar curves.

r=e^c

note c=theta

I cant seem to get the right answer. Also, I need help understanding the differences between dy/dx, dy/dc, dr/dc, dx/dc, etc. when you are talking about polar coordinates.

I need to find the horizontal and Vertical tangent lines

Answer 1

This function will produce a spiral as r will always increase as theta increases. (I much prefer to use t to stand for theta.)
To find the gradient of the curve and then the equation of the tangent you need to find dy/dx. You start this from the conversion equations
x = r*cos(t) and y = r*sin(t)
from which you can find dx/dt and dy/dt and then you can use dy/dx = (dy/dt)/(dx/dt). In this case
x = r*cos(t) = e^t*cos(t) ---> dx/dt = e^t*cos(t) - e^t*sin(t)
y = r*sin(t) = e^t*sin(t) ---> dy/dt = e^t*sin(t) + e^t*cos(t)
Can you finish it from there?
Horizontal tangents will occur when dy/dt = 0 and vertical tangents will occur when dx/dt = 0.

Answer 2

r=e^@
x=r cos @
y=r sin @ so
x= e^@ cos @
and y=e^@ sin @
dy/d@= e^@sin @+e@ cos @
dx/d@= e^@cos@-e^@ sin @

so dy/dx =( sin@+cos@)/(cos@-sin @)

In general
dx/d@= dr/d@*cos@-r sin@
dy/d@= dr/d@*sin@+r cos@ and dividing the second by the first you get
dy/dx.