a is the amplitube
b is the frequency
c is the horizontal shift
d is the vertical shift.
formula for frequency is f = 360 / p
f is the frequency
p is the period
if you want one full sine wave to be 60 seconds, then let 60 degrees represent 60 seconds.
your period is 60 seconds.
f = 360 / 60 = 6
if your frequency is 6, then you will get 6 full sine waves in 360 degrees.
that would make each full sine wave equal to 60 degrees.
with b = 6, your sine wave function becomes y = a * sin(6 * (x - c)) + d
when a = 1, the sine wave is plus or minus 1 unit from the center line of the sine wave.
that makes the top 1 unit above the center line and the bottom 1 unit below the center line.
the center line of the sine wave is the x-axis without any adjustment.
if you let the x-axis represent the ground and you want the bottom of the sine wave to be .5 units above the ground, then d must be equal to 1.5
that raises the center line of the sine wave to y = 1.5 which means the bottom of the sine wave will be at y = .5 which is .5 units above the ground.
set d to 1.5 and the formula becomes y = a * sin(6 * (x - c)) + 1.5
let's take a look at whaat we have so far.
since c is normally 0, we set it to 0.
since a is normall 1, we set it to 1.
the function becomes y = 1 * sin(6 * (x - 0)) + 1.5
this is what it looks like on the graph.
the sine wave starts at the center line which is 1.5 units above the x-axis which represents the ground.
we want the sine wave to start at .5 units above the ground.
from the graph we can see that if we shift the sine wave to the right 15 degrees, then it will start .5 units above the ground.
we need a horizontal shift to the right of 15 degrees.
shift to the right means c must be positive.
therefore we make c = 15 and the formula becomes y = 1 * sin(6 * (x - 15)) + 1.5
this is what that looks like on the graph.
the final diagram is what you want.
one full cycle is 60 degrees which represents 60 seconds.
the cycle starts .5 meters above the ground.
the a point on the ferris wheel has a minimum height of .5 meters above the ground and a maximum height of 2.5 meters above the ground.
the amplitude of the sine plus or minus 1 unit from the center line which is at 1.5 meters above the ground.
the point on the ferris wheel that was .5 meters from the ground when it starts will be back to .5 meters above the ground every 60 seconds.
here's a view of what the sine wave looks like after it went through three 60 second cycles.
your final equation was y = 1 * sin(6 * (x-15)) + 1.5
since a = 1 is default, the equation can be written as y = sin(6 * (x-15)) + 1.5
here's a reference on the equation to graph a sine wave.
https://mathbitsnotebook.com/Algebra2/TrigGraphs/TGsinusoidal.html