A point P is located on the rim of a giant wheel of radius 40m centred at the origin of the coordinate plane. At time t=0, the point P is at (40,0) and the wheel starts rotating counter-clockwise. After completing 125 full revolutions it immediately rotates further through an angle of −419π (clockwise).
(a) Express the total angular displacement (in radians) experienced by P as a single angle.
(b) Reduce the angle obtained in part (a) to its principal value lying between 0 and 2π.
(c) Using the principal value, determine the exact coordinates of P after the described motion.
(d) Hence find the vertical height of P above the centre of the wheel at this final position.