Question

The plane curve is given by the equation R(t)= ( ln sin t)i + ( ln cos t)j.
find the unit normal vector to the curve t= ¤Ð/6.

Answer 1

assuming you meant t=π/6

The unit tangent vector
T(t) = r'/|r'| =

cot(t) i - tan(t) j
-------------------------
tan^2(t) + cot^2(t)

The unit normal is T'/|T'| =

-(csc^2(t) i + sec^2(t) j)
----------------------------------
csc^4(t) + sec^4(t)

which at t = π/6 is

-(4i + 4/3 j)/(16 + 16/9)
= -4/3 (3i+j) * 9/156
= 1/13 (3i+j)

as always, check my math