# Thread: Determining equations for secant/cosecant curves.

1. ## Determining equations for secant/cosecant curves.

I have a question on my homework that I don't really understand, and the textbook isn't helping me at all, I was hoping any of you could. I was ill and still haven't gotten the notes I've missed, but the test is 2 days away

Can someone tell me the steps I need to do in order to figure this out? I'm really having trouble with it.

2. Hello Random-Hero-
Originally Posted by Random-Hero-
I have a question on my homework that I don't really understand, and the textbook isn't helping me at all, I was hoping any of you could. I was ill and still haven't gotten the notes I've missed, but the test is 2 days away

Can someone tell me the steps I need to do in order to figure this out? I'm really having trouble with it.

This is what you need to know:

(1) $\displaystyle \sec(x) = \frac{1}{\cos(x)}$

(2) $\displaystyle \cos (x) =0$ when $\displaystyle x = \frac{\pi}{2}, \frac{3\pi}{2},\frac{5\pi}{2},...$

So for these values of $\displaystyle x, \sec(x)$ does not exist.

(3) The maximum value of $\displaystyle \cos(x)$ is $\displaystyle 1$, and this occurs when $\displaystyle x = 0, 2\pi, 4\pi, ...$

So the minimum positive value of $\displaystyle \sec(x)$ is $\displaystyle 1$ at these values of $\displaystyle x$.

Similarly, the maximum negative value of $\displaystyle \sec(x)$ is $\displaystyle -1$ at $\displaystyle x = \pi, 3\pi, 5\pi, ...$

(a) Now let's suppose that the function to give the graph is of the form $\displaystyle y = a\sec(bx+c)$

Using (3) above, the minimum positive value of $\displaystyle y$ is $\displaystyle a$. So, from the graph, $\displaystyle a = 3$.

The first of these is when $\displaystyle x = \frac{\pi}{4}$. So from (3) again, $\displaystyle \frac{b\pi}{4}+ c = 0$

The next is when $\displaystyle x = \frac{3\pi}{4}$. So $\displaystyle \frac{3b\pi}{4}+ c = 2\pi$

Subtracting these two equations, we get $\displaystyle \Big(\frac{3b\pi}{4} +c\Big) - \Big( \frac{b\pi}{4}+ c\Big) = 2\pi - 0$

$\displaystyle \Rightarrow \frac{2b\pi}{4}=2\pi$

$\displaystyle \Rightarrow b =4$

$\displaystyle \Rightarrow c = -\pi$

So the function is $\displaystyle y = 3\sec(4x-\pi)$

(b) Now start with the fact that $\displaystyle \csc(x) = \frac{1}{\sin(x)}$ and see if you can work out similar statements to numbers (1) - (3) above. Then set up the function $\displaystyle y = a\csc(bx+c)$ and work out the values of $\displaystyle a, b, c$ in the same way.