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Math Help - Finding all of the angles when sec(3x)

  1. #1
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    Thumbs up 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!
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  2. #2
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    Re: Finding all of the angles when sec(3x)

    Quote Originally Posted by Zellator View Post
    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.
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  3. #3
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    pickslides's Avatar
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    Re: Finding all of the angles when sec(3x)

    Hi Zellator, you have the right idea

    Quote Originally Posted by Zellator View Post

    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.
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  4. #4
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    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
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