Mathematics

Problem

For

0\le t\le \dfrac{\pi}{3}

, a curve is given by

x(t)=k\sin t,\qquad y(t)=k\cos t,

where

k

is a positive constant to be determined. a. Express the arc length

S(k)

of the curve from

t=0

t=\dfrac{\pi}{3}

in terms of

k

. Do not evaluate numerical constants yet. b. The arc length is known to be

\dfrac{8\pi}{3}

. Solve for

k

. c. Using the value of

k

found in part (b), give the exact coordinates of the endpoint of the curve at

t=\dfrac{\pi}{3}