Write a sine and a cosine equation to model the movement of the boat in the following case:A boat tied at a dock moves up and down with the passing...

We know that any periodic motion is a simple harmonic
motion (SHM) and could be written as x(t) = A sin (wt+p), where A is the amplitude , w
is the angular velocity, p is the phase difference and t is the time in
minutes.

Since the vertical maximum distance between the
highest and lowest points in the motion of the boat  is 9 m, the amplitude A = 9/2 = 4.5
m

So here,  A = 9/2, w = 2pi/ 5 radians per
minute.

So at the time t = 0, the equation of motion given
by : -4.5 = 4.5 sin {5t+p}. So sin (5*0+p) =
-1.

=> 5t+p = -pi/2, or p =
-pi/2.

Therefore the required model of the motion is x(t) =
4.5 sin(5t-pi/2) in terms of sin function.

To write the
equation in terms of Cosine  function:

Since cos x = sin
(pi/2 - x), we can rewrite the above equation as
below:

x(t) = 4.5 cos {pi/2 - (
5t-pi/2)}.

=>  x(t) = 4.5 cos {-5wt +
pi}.

=> x(t) = 4.5 cos (5wt- pi) , as cos (-x) =
cosx.

Therefore the required equation of the simple
harmonic motion of the boat is given by:

x(t) = 4.5
(5wt-pi). Or x (t) = cos (5t+pi) , as x(t) = x(t+2pi) for any
SHM.