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

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

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

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

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

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

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

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

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

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

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

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

$\Rightarrow b =4$

$\Rightarrow c = -\pi$

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

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