Solving for x in a sinusoidal function

Hello,

I have a homework question that asks to solve for x (which is time in seconds) in the sinusoidal function: * y=4sin45x + 5, where y = 1*

The answer is supposed to be 38, but I get 2 like this:

1 = 4sin45x + 5

-4 =4sin45x

-1 =sin45x

sin^{-1}(-1) = 45x

-90 = 45x

x=-2

What have I done incorrectly? Thanks for your help

Re: Solving for x in a sinusoidal function

You got it: $\displaystyle x = -2$ is correct, sort of. The answer to your question has to do with the limited range of the inverse sine function.

The sine function in your problem has a period of 8 seconds, so whatever phenomenon you are investigating repeats every 8 seconds. It is at a position where $\displaystyle y = 1$ when $\displaystyle x = -2 \pm 8$, so this is the solution to your equation. In order to answer your question you would have to be given more information as to which time is correct. For example, the problem statement may say to "look for the first solution after a half-minute".

Re: Solving for x in a sinusoidal function

And, of course, assuming that "45x" is in degrees. That is not stated in your problem, and is a bit unusual (in problems where the argument of the trig function is not specifically an angle, it is more often in radians) but since "90" works so nicely here, that is probably true!

Re: Solving for x in a sinusoidal function

Maybe I didn't understand the wording: (Background info: two seconds into the ferris wheel ride, he is 9m above the ground. This is the highest point he will reach. the lowest point he will reach is 1m above the ground. The period of the function is 8s)

One question (#1)asked. " Using your knowledge of periodic functions, explain how you would be able to tell how high the person is off the ground (in a ferris wheel) after he has been on the ride for 38s. Is it safe for him to get off if the wheel were stopped? I calculated this to be 1m by substituting 38 in for x.

The next question (#2)asked. " Determine the equation of the sinusoidal function that expresses his height above the ground in terms of time" Well here I thought we already had that equation.

Then #3 asked, "Use your equation from #2 to verify your answer to #1.

What am I missing here?

Re: Solving for x in a sinusoidal function

$\displaystyle \sin(45^\circ x) = -1$ when $\displaystyle x = -2 + 8k$ for any $\displaystyle k \in \mathbb{Z}$. So, you need to verify that there exists a $\displaystyle k \in \mathbb{Z}$ such that $\displaystyle 38 = -2+8k$. In fact, there is. $\displaystyle k=5$.

Re: Solving for x in a sinusoidal function

Essentially this is exactly what is going on. According to your equation, he will be at the minimum height of 1m after 6, 14, 22, 30, and 38 seconds. And more after that too unless he stops. My apologies for previously saying the actual solution is $\displaystyle -2 \pm 8$. I meant to say the solution is $\displaystyle -2 \pm 8n$, where $\displaystyle n$ is any integer.

I suspect the difference between the problems is that in #1 you should think graphically (or even just logic it out) and in #2 you should think analytically (create the equation). In #3 you are asked to verify your thoughts from #1 using the equation (which is what you're asking about here). An easier way to do it might be to simply plug 38sec in for the time and see that the height will be 1m when you do this.

And I've got to reiterate what HallsofIvy says here too: I would not use degrees in this situation. The equation should be written using radians: $\displaystyle h(t) = 4sin(\frac{\pi}{4}t)+5$ unless you're very specific about the use of degrees.

Re: Solving for x in a sinusoidal function