LogFAQs > #959489220

That isn't an arcsine graph. What you have in this second example is a linear graph made periodic by taking the inverse sine of a sine. If that doesn't make sense, consider the following line:

A(x) = 3 sin[x] --> A(x)/3 = sin[x] --> y(x) = arcsin[A(x)/3] = arcsin[sin[x]] = x --> y(x)=x

The slope switches direction because the computation is switching signs at pi/2, 3pi/2, etc.

An arcsine graph should be curved, and to be a function, you must restrict the domain to an appropriate interval. The domain of the arcsine function will simply be the range of the corresponding sine function, and vice-versa. Since you're looking for Q4 and Q1 data, the most straightforward domain for the sine function would then be -pi/2 < x < pi/2. In the case above, this corresponds to a range of -3 < y < 3. Switching the variables will give the domain of the arcsine function, then, as -3 < x < 3. Finally, you need to be careful about how you set up the arcsine function, as the total argument (or input) can only accept values between -1 and 1. This is because the output of a sine operator is only ever between -1 and 1, so this is the only acceptable input to an arcsine. Thus, in this example, the arcsine function should look like this:

y(x) = sin^-1 [x/3] , -3 < x < 3

Plug this in and you'll get a graph that looks much different.

The same logic applies to the problem at hand. The first equation looks fine except for the argument in the sine function. I say this because you identified the period as 0.9, but used this value where the angular frequency should be. This is an easy fix, as the period T is related to the angular frequency w by the relation:

w = 2pi/T

whereas the general form of an oscillating function (without phase) is:

f(t) = A sin[wt] + C , C = constant

Of course, you can swap out the variables for the purpose of plotting this without changing anything. The addition of the constant 135 at the tail end translates the graph up the y-axis to the chosen midline, so no problem there. Just note that you cannot include this constant in the argument of the arcsine function, for the reason already discussed.

So, changing our variables and rearranging, we have:

y = A sin[wx] + C --> (y - C)/A = sin[wx] --> wx = sin^-1 [(y-C)/A] --> x = (1/w) sin^-1 [(y-C)/A]

Once you plug in your values, swap the variables for the purpose of plotting, and define an appropriate domain, you should be good to go. Note that this will be centered about the origin, so you'll need to add a constant if you wish to translate it such that both graphs share a midline.

Hope that's helpful.

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