The vertical location of the tip of a clock’s second hand (y) relative to the clock’s center measured in centimetres is modelled by the equation y= 3cos (πs/30), where s is the number of seconds that have passed.

Question

a) How long is the second hand, and where does it start?
b) Use transformations to explain how you could change the original equation to find the vertical location of the tip of the hour and minute hands in terms of y and s if the hour hand is 1 centimetre long and the minute hand is 4 centimetres long.
c) What is a simple way you could change the equations to represent the hands’ horizontal positions? Why does this work?

Answer 1

(a) 3
(b) change the period. Note that cos(2ks) has a period 1/2 as long as cos(ks)
(c) cos(k(s-θ)) is shifted later by θ