A sinusoidal function is of the form \(\displaystyle{h}={a}{\sin{{\left[{b}{\left({t}−{c}\right)}\right]}}}+{d}{\quad\text{or}\quad}{h}={a}{\cos{{\left[{b}{\left({t}−{c}\right)}\right]}}}+{d}\) where ∣a∣∣a∣ is the amplitude, \(2π/b\) is the period, c is the phase shift, and dd is the vertical shift (its midline).

The diameter of the wheel is twice the amplitude. Since the diameter of the wheel is 90 ft, then the amplitude is \(∣a∣=90/2=45.\)

The axle point of the wheel is the vertical shift of the sinusoid. Since the axle of the wheel will be 55 ft above the ground, then d=55.

The wheel makes one revolution every 60 seconds so the period is 60. Therefore \(2π/b=60\). Solving this for b gives \(b=2π/60=π/30\).

A since curve starts at its midline and a cosine curve starts at its maximum. Since we need the function to start at the ground, which is the minimum of the function, we need to use the cosine form with \(a<0\) or use a sine curve that has a phase shift.

If we use a cosine curve, then \(a=−45\) so it will start at the minimum and we don't need a phase shift. We then have everything we need to write the cosine function. Substituting \(a=−45\), \(b=π/30\), \(c=0\), and \(d=55\) into \(h=a \cos[b(t−c)]+d\) then gives \(h=−45cos(π/30)t+55\).

If we use a sine curve, we need to determine how much the phase shift needs to be so that it will start at a minimum. \(y=\sin x\) has a minimum at \((−π2,−1)\). Since \(b=π/30\), then the parent function has been horizontally compressed by a factor of \(30/π\). The minimum after the compression is then \((−(\pi/2)⋅30/π,−1)=(−15,−1)\). The graph after the compression must then be horizontally translated right 15 units to get a minimum at (0,−1). Therefore \(c=15\). Note that the minimum of our graph is not (0,−1), we just needed to determine how far horizontally the graph needed to be moved, which is not affected by the change in amplitude and vertical shift.

Since we don't need a reflection, then \(a=45\) for the sine curve. We then have everything we need to write the sine function. Substituting \(a=45, b=π/30, c=15\) and d=55 into \(h=a\sin[b(t−c)]+d\) then gives \(h=45\sin[π30(t−15)]+55\).

Part 2:

A piece of evidence that sine and cosine are basically the same thing is that they have the same domain of all real numbers and range of [−1,1].

A piece of evidence that they are not the same thing, is that they have different starting points (the point where x=0) and different intervals of increasing and decreasing. Sine has a starting point at the origin, increases to a maximum at \(x=π/2\), decreases to a minimum at \(x=3π/2\), increases to a maximum at \(x=5π/2\), and continues this pattern of increasing/decreasing every \(\pi\) units. Cosine, however, starts at a maximum at \(x=0\), decreasing to a minimum and \(x=π\), increases to a maximum at \(x=2π\), and continues this pattern of increasing/decreasing every π units.