text: # 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
explanation: The document is a forum thread discussing a math problem. It contains a question about finding the sum of all solutions in the interval \([0, 2\pi)\) for the equation \(\sec(3x) = \sqrt{2}\). The thread also includes responses that provide guidance and confirm the user's approach.
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distilabel_metadata.content: # 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
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a forum thread discussing a math problem. It contains a question about finding the sum of all solutions in the interval \([0, 2\pi)\) for the equation \(\sec(3x) = \sqrt{2}\). The thread also includes responses that provide guidance and confirm the user's approach.
  
  Label: question-and-answer