# Thread: Finding all of the angles when sec(3x)

1. ## Finding all of the angles when sec(3x)

Hi Forum
I have some ideas myself on this one, but I'd love to hear how you guys would proceed in this scenario.

Find the sum of all solutions in the interval $\displaystyle [0,2\pi)$ for the equation $\displaystyle sec(3x)=\sqrt2$

First, we can put this in the form $\displaystyle cos(3x)=\frac{1}{\sqrt2}$ for better viewing.
We know that $\displaystyle cos(x)=\frac{1}{\sqrt2}$ is $\displaystyle \frac{\pi}{4}$ and $\displaystyle \frac{7\pi}{4}$.

Then our period will equal $\displaystyle \frac{2\pi}{3}$, so the distance from one $\displaystyle \sqrt2$ to another is $\displaystyle \frac{\pi}{3}$
Since our $\displaystyle \sqrt2$ will appear 3 times faster because of the $\displaystyle 3x$, our first $\displaystyle \sqrt2$ is $\displaystyle \frac{\pi}{12}$

Now, can we just go summing it to $\displaystyle \frac{\pi}{3}$ until we get to $\displaystyle 2\pi$?
What do you guys think?

Thanks!

2. ## Re: Finding all of the angles when sec(3x)

Originally Posted by Zellator
Hi Forum
I have some ideas myself on this one, but I'd love to hear how you guys would proceed in this scenario.

Find the sum of all solutions in the interval $\displaystyle [0,2\pi)$ for the equation $\displaystyle sec(3x)=\sqrt2$

First, we can put this in the form $\displaystyle cos(3x)=\frac{1}{\sqrt2}$ for better viewing.
We know that $\displaystyle cos(x)=\frac{1}{\sqrt2}$ is $\displaystyle \frac{\pi}{4}$ and $\displaystyle \frac{7\pi}{4}$.

Then our period will equal $\displaystyle \frac{2\pi}{3}$, so the distance from one $\displaystyle \sqrt2$ to another is $\displaystyle \frac{\pi}{3}$
Since our $\displaystyle \sqrt2$ will appear 3 times faster because of the $\displaystyle 3x$, our first $\displaystyle \sqrt2$ is $\displaystyle \frac{\pi}{12}$

Now, can we just go summing it to $\displaystyle \frac{\pi}{3}$ until we get to $\displaystyle 2\pi$?
What do you guys think?

Thanks!
I think you should find all the values of x that lie in the given interval and then add them up.

3. ## Re: Finding all of the angles when sec(3x)

Hi Zellator, you have the right idea

Originally Posted by Zellator

First, we can put this in the form $\displaystyle cos(3x)=\frac{1}{\sqrt2}$ for better viewing.
We know that $\displaystyle cos(x)=\frac{1}{\sqrt2}$ is $\displaystyle \frac{\pi}{4}$ and $\displaystyle \frac{7\pi}{4}$.
$\displaystyle \cos(3x)=\frac{1}{\sqrt2}\implies 3x = \frac{\pi}{4},\frac{7\pi}{4}\implies x = \frac{\pi}{12},\frac{7\pi}{12}$

Now add $\displaystyle \frac{2\pi}{3}$ to these solutions, up to $\displaystyle 2\pi$

After that add them up as suggested in post #2.

4. ## Re: Finding all of the angles when sec(3x)

Hey mr fantastic and pickslides!

It's great to know that I am on the right track, then.
Thanks for your second opinions, that was really appreciated