# 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 $[0,2\pi)$ for the equation $sec(3x)=\sqrt2$

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

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

Now, can we just go summing it to $\frac{\pi}{3}$ until we get to $2\pi$?
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 $[0,2\pi)$ for the equation $sec(3x)=\sqrt2$

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

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

Now, can we just go summing it to $\frac{\pi}{3}$ until we get to $2\pi$?
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 $cos(3x)=\frac{1}{\sqrt2}$ for better viewing.
We know that $cos(x)=\frac{1}{\sqrt2}$ is $\frac{\pi}{4}$ and $\frac{7\pi}{4}$.
$\cos(3x)=\frac{1}{\sqrt2}\implies 3x = \frac{\pi}{4},\frac{7\pi}{4}\implies x = \frac{\pi}{12},\frac{7\pi}{12}$

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

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
The text contains a mathematical question and a discussion on how to solve it, but no explicit answer to the sum of all solutions is provided.
single_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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 $[0,2\pi)$ for the equation $sec(3x)=\sqrt2$

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

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

Now, can we just go summing it to $\frac{\pi}{3}$ until we get to $2\pi$?
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 $[0,2\pi)$ for the equation $sec(3x)=\sqrt2$

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

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

Now, can we just go summing it to $\frac{\pi}{3}$ until we get to $2\pi$?
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 $cos(3x)=\frac{1}{\sqrt2}$ for better viewing.
We know that $cos(x)=\frac{1}{\sqrt2}$ is $\frac{\pi}{4}$ and $\frac{7\pi}{4}$.
$\cos(3x)=\frac{1}{\sqrt2}\implies 3x = \frac{\pi}{4},\frac{7\pi}{4}\implies x = \frac{\pi}{12},\frac{7\pi}{12}$

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

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
user
The text contains a mathematical question and a discussion on how to solve it, but no explicit answer to the sum of all solutions is provided.

TYPE: SINGLE_QUESTION