1 cycle is 360 degrees.

2pi radians is 360 degrees.

1 cycle = 2pi radians

4 cycles/second(2pi/cycle) = 8 pi radians /second

You are transforming from the x domain to the t domain.

y=a sin k (x-b) +c

At x=0, y = asin0 = 25(1), so a=25

at t=0, y=25f(t) = 25f(0)

at t= 1/4 seconds (one full cycle), y = 25, x =2pi

at t = 1/8 seconds, (one half cycle), y = -25, x = pi

at t = 1/16 seconds, (one quarter cycle), y =0, x=pi/2

...

f(t) = sin k(x-b) = sin 8pi(x-b)  (I'm going to dispense with c since there is no phase offset or damping)

The questions you posted threw a curve in by asking for an equation with the motion starting at the minimum deflection, hence the cosine term.

The equation for simple harmonic motion you posted is also missing a pair of parentheses in the sin function.

I'm curious about whether this was for a physics, dynamics, or calculus class.

@setlab, in case you haven't revisited your question.

This should help:

http://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/TrigFunctions.aspx

I would have done this a little different if I had known this was for a trig class.  I'll post the solution later in a format that will be clear, that equation with "t" won't appear like a miracle.

Try this, first some illustrations of three possible solutions; you can see that solution 3 satisfies the problem statement -



Then some illustrations for clarification of the problem -





Then a tabulated example, and the solution -



The key in this solution is to calculate the period, then calculate the angle at a particular time, t.  Notice that t/T is simply the fraction of a full cycle at t.  That's all there is to the problem.

To calculate the solution for the phase change at the end of the problem, y = -25cos(2pi(t/T) + C), where C is the phase change.

Frankly, I think the problem is a little advanced at that point.  The main value of the exercise is to find out whether the student knows the shape of sine and cosine functions, their values at 0 degrees, 90 degrees, and so on, and the sign of the functions in each quadrant of the circle.

On the other hand, if you can reproduce a table similar to the one I show above, calculate the frequency and period, and understand how I arrived at the solution, I'll guess you will be in the top 2% of a trig class.

I mostly do analysis of air vehicle structure.  What I like to do at work is teach or coach, and invent.  I always recommend returning to the fundamentals to solve problems.

A spreadsheet would be easy to put together, but the entries would look a little different.  I thought about making something up to plot the functions, but didn't really have time so I borrowed a couple of useful plots I found on line.

I doubt the thread author has seen undamped or damped harmonic motion or a Laplace transform at this time.  The problem posed is a little advanced for a trig class, but it's a good way to probe understanding.

Besides that, the problem given is clearly not damped.