Hi,

I am studying trigonometry on my own. I am blind, so I'm using audio books. Since I'm blind, please do not provide any type of graphics or pictures in your response.

The issue here has to do with asymptotes of secant and coSecant functions. Here's the latest problem for which my answer is different from what the audio book reader says:

I'm typing the problem exactly as the reader reads it.
y = 2 secant of the difference 2x minus pi for x greater than or equal to negative 2 pi and less than or equal to 2 pi

No issue with the amplitude -- it's 2.
No issue, unless I'm really lost, with the period -- the period is pi.
The phase shift is pi over 2 to the right.
The reader and I even agree that the y intercept is at a y value of negative 2 with a downward opening parabola.

So, what's the issue?
The reader says that the asymptotes are at pi over 8, 3 pi over 8, 5 pi over 8, and 7 pi over 8. (With corresponding negative values to the left of the y axis.)
I think that the asymptotes are at pi over 4, 3 pi over 4, 5 pi over 4, and 7 pi over 4. (With corresponding negative values to the left of the y axis.)
My thinking is that the asymptotes are at the same points as where the coSine graph intersects the x axis. And since the period is pi ...

Most of the time, my answers agree with the reader's. But sometimes, the book has errors (the readers are instructed by the company they work for to read exactly what's in the books; including errors. Sometimes the reader misspeaks -- for example, in this problem, he says that the low point of each upward opening parabola is negative 2 (when it is clearly 2).

I do this all in my head. First, I imagine the coSine (or sine) function, including the amplitude. Then, I do any phase shift. Then, in the case of secant and coSecant functions, I imagine those curves.
If I'm wrong about the asymptotes, please explain what's wrong with my approach.

Thanks

 

I'm not sure what "amplitude" even means for a secant waveform.

I agree that the period is pi.

When you talk about phase shift, that needs to be relative to some reference. What reference are you using?

Yes, the vertical asymptotes are going to be where the cosine of 2x minus pi goes to zero and that gives the values that you come up with, so either the book or the reader was wrong (my guess is the book, it doesn't seem like the kind of mistake a reader would make). My best guess is that this problem was a mod of a similar problem that used 4x as the frequency and the range from -pi to +pi and the person doing the editing didn't update it completely.