Question

if given a certain value how would I find what angle does the unit tangent make with the x-axis?

Answer 1

that would be dy/dx

Answer 2

so my unit tangent vector T(w) = cos(w)i+sin(w)j and say w=-pi/3 how would i find what angle does the unit tangent vector T(w) make with the x-axis?

Answer 3

since you have parametric equations,
y = sinw
x = cosw
dy/dx = (dy/dw) / (dx/dw) = cosw/-sinw = -cotw

or note that the equation could have been written
x^2 + y^2 = 1
x + y y' = 0
y' = -x/y = -cosw/sinw = -cotw

Answer 4

and then i would plug in my value for w to find the angle?

Answer 5

oops. sorry. I didn't notice that T was the tangent vector. I took it to be the original position vector. So you must have had
r(w) = sinw i - cosw j
so things are backwards. But you can see how to do it.

Answer 6

my r(w) = wi-ln(cosw)j so r'(w) = i + tanwj. so to find the angle it would be tan(w)