Ughhhh, why do I always make mistakes when writing my questions on AN??

Yep! I meant 'find h if d=0 when t=0', whoops! The actual question is much longer, but I'll include the answers from the previous questions for extra information. All I really know is that d is the distance from the water surface to point P on a water wheel with radius 3, and t is the time is seconds. Also, the wheel rotates at 4 revolutions per minute.

*What is h if d=0 when t=0 for the equation d= 2 +bsin(2π/15(t-h))*. For this equation, the period is 15 and the amplitude is 3. I put 'b' because we don't know the exact value, right? (It's either 3 or -3. Correct me if I'm wrong I'm just confused on whether the value of h is indefinite, so any explanation on why h = 1.74 would be great Thank you for your help

All good! At least you're honest about it - I've had some students (not from AN) come to me and go, "no!! Of course I copied the question right, maybe you're just a bad tutor?!", and then when I gave them proof that the question they asked couldn't be answered, they'd come back with, "oh, turns out I copied it wrong. It's actually this" without even an apology.

Okay, so now that I've got a better understanding of what's going on, let's work with what we know. d=0 when t=0, this gives us:

Since h is a translation factor, we know that it can have an INFINITE possible number of values - but all of them should be spaced 15 seconds apart. The reason it's 15 seconds apart is because the period is 15 seconds (feel free to confirm this for yourself). So, let's find the first value of h for 0<t<15, so first we solve for h:

Okay, so this is immediately frustrating because we can't use exact values, and don't know if b is negative or positive!!. But, that's not as difficult as you think. Here, consider the two sin waves below:

https://www.desmos.com/calculator/9cvmthhh0u (note that you can turn off graphs by clicking the red or blue circle)

So, there is logic I could use to solve this. Notice that the red curve immediately goes up, while the right curve immediately goes down. That would mean that the sign of b is going to depend on whether that point P initially goes up or down. But that logic will rely on information you don't necessarily know - so instead, let's just be simple, and assume b=+3. If we do that, we get:

Which is right! Note that in the third line, we didn't need to worry about solving for a specific domain, because we only care about the first possible value that h can be in this instance. For example, let's say you were instead interested in a probability question. If I asked you what's the probability of drawing an ace, it's going to be 4/52=1/13. Now, what if I asked yo uwhat's the probability of drawing an ace if you use TWO decks? You might immediately jump on it being 8/104 - but the trick is, the answer is still 1/13. And it's going to be 1/13 no matter how many decks you add, because the amount of cards being added matches the ratio of aces being added. It's the same idea here - because sine waves are periodic, it doesn't matter which h you pick, because they won't affect what the graph looks like at the end. (and if it made sense to you before I used the card analogy, and the card analogy just confused you, simply ignore it and stick with your initial instincts)

But if that doesn't sit well with you, it shouldn't - it doesn't with me, either. So, let's see what happens if we let b=-3:

How weird - we get the exact same number, just a different sign! If you're curious, that's because sin(x) is an odd function (f(-x)=-f(x), I think this definition unfortunately got removed from the methods curriculum, but it's still kinda cool!), and doesn't really have bearing on this question. So, how do our graphs look with the h value added in?

https://www.desmos.com/calculator/sbrggivf0kWell, the truth is, they both have d=0 when t=0, so as far as I'm concerned - both are valid answers. See your teacher for any reason as to why one of these answers isn't correct - it might be due to some information that you didn't even realise was information!

h is that value since if you shift it, it each minute it begins and ends at y = 0 after 4 rotations.

If you graph the funciton you'll notice.

*(Image removed from quote.)*

That at 0 seconds and 60 seconds it is at y = 0, I'm assuming that's the reason for the shift is there any part of the quesetion that states that d starts at 0?

If so, t = 0 and d = 0 that is (0, 0). There are technically infinte solutions yes. That's why people write it as a function like cos (45 + 360k) where k is an integer. But they probably chose the closest solution or something. It's basically asking 'for what value of h does the function intersect (0, 0)'

The example cos (45 + 360k) was made up btw.

You're not wrong, the problem is your answer misses the question that was being asked. In your graph, you've chosen there are two points that intersect with the x-axis - one in which the curve then starts going down, and one in which the curve starts going up. We don't know which of these points is making the intersection in Azila's answer, hence why I've explored what happens for both cases.