Trig: Writing the equation for vertical asymptote of a secant function?

[DinosaurEgg]

Homework Statement

Homework Equations

How did they come up with \frac{1}{2}+k for the equation of the vertical asymptote? I understand everything else except this.

The Attempt at a Solution

On this particular exercise, I graphed it and saw that each of my vertical dashed lines were all one whole unit apart. I've tried this method with another problem that had the dashed lines separated 2 units apart, took that 2, and multiplied it by the x=\prod/2 + k\prod. The result was \prod+2k\prod, which was correct.

I tried it with other numbers and have gotten the correct answer, but I have a feeling I'm still doing something wrong. Because with this particular one using my method, I input \prod/2 + k\prod as my answer for the vertical asymptote which was incorrect. What am I doing wrong?

 

[Dick]

You would be right if the equation were y=4*sec(x). It's not. It's y=4*sec(pi*x). There's already a pi in the equation for y. y=4*sec(pi*x) doesn't have an asymptote at x=pi/2.

 

[DinosaurEgg]

I noticed multiplying the vertical asymptote formula/equation by 1/pi cancels out the pi, resulting in that 1/2+k... but where did they get 1/pi from? Does that have any relation to secant being 1/cos?

 

[DinosaurEgg]

Worked on another problem set up similarly and I think I got it!

I noticed that simply taking the 'B' (like in the y=Asin[B(x-C)]+D formula), turning it into the reciprocal (1/B), and thennnn multiplying it by \prod/2 + k\prod gets me the right asymptote. This *does* relate to inverse trig functions (ie, sec being the reciprocal of cos), right?