Find the slope of the tangent line to the given polar curve at the point specified by the value of Ɵ using the exact answer: r = 2 - sinƟ, Ɵ = π/3?

tangent-line
polar-curve-point

#1

Find the slope of the tangent line to the given polar curve at the point specified by the value of Ɵ using the exact answer: r = 2 - sinƟ, Ɵ = π/3 ?

Answer:

In polar coordinates, we know that
x=rcost
y=rsint

Then dx/dt=-rsint and dy/dt=rcost,
(dy/dt)/(dx/dt)=dy/dx=(-rsint)/(rcost)=-cott
since we want to find y’ at the pint t=pi/3 then our answer is
-cot(pi/3)=-1/(sqrt(3))