The equation of the water level y, is approximately y=Asin(nt+b).

Question

a) amplitude, A. (1.91m)
b) what is the water level at midnight? (0.95m)
c) Find b to two decimal places.
d) Find n to one decimal place.
1 full wavelength is 12.37hours (given), so 2pi/12.3 = 0.5

At t=0, y=0.95
For C, I can't seem to get the right answer, which is 0.52

Answer 1

So far, so good. You have
y = 1.91 sin(0.5x + b)
But, you want y(0) = 0.95, so
1.91 sin(b) = 0.95
sin(b) = 0.5
b = π/6 = 0.5235

Answer 2

"b) what is the water level at midnight? (0.95m)" is confusing
Is the water level at midnight 0.95 ???
I will assume that is what is meant.

amplitude is 1.91 , so
y= 1.91sin(nt+b)

1 full wavelength is 12.37hours (given), period = 2π/n
2π/n = 12.37
n = 2π/12.37 = .508 , so
y = 1.91sin (.508t + b)

at midnight, t=0, and y = .95
.95 = 1.91sin(.508t + b)
sin(.508(0) + b) = .497382...
b = .52057...

so y = 1.91sin(.508t + .52057)

This is a confusing and poorly worded question.

Answer 3

y = A sin (2 pi t/T + b)
T given as 12.37 ? so 2 pi/T = your n = 0.508
then 2 pi/12.37 = .508
y = 1.91 sin ( .508 t +b)
at t = 0, y = 0.95
0.95 = 1.91 sin b
sin b = 0.497
b = 30 degrees close enough = pi/6 radians = 0.523 radians
(So I agree with your 0.52
now y = 1.91 sin (.508 t + .523)