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Thread: cot(x) + 3cot(2x) - 1 = 0

  1. #1
    Member Furyan's Avatar
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    cot(x) + 3cot(2x) - 1 = 0

    Hello this question looked straight forward enough, but I'm stuck and would really appreciate some help.

    Solve on the interval $\displaystyle 0\leq x \leq\2\pi$

    $\displaystyle \cot\theta + 3\cot2\theta-1=0$

    I tried writing it in terms of $\displaystyle \cos\theta$ and $\displaystyle \sin\theta$

    And got this;

    $\displaystyle \dfrac{5\cos^2\theat - 3\sin^2\theta}{2\sin\theta\cos\theta} =1 $, among various similar equations.

    I have worked on from here a bit, but can't seem to get it in terms of a trigonometric equation I can then solve. I'd really appreciate it if somone would tell me if I'm on the right track and should keep going or if I need to approach the question in a different way.

    Thank you very much.
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  2. #2
    Super Member ILikeSerena's Avatar
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    Re: cot(x) + 3cot(2x) - 1 = 0

    Quote Originally Posted by Furyan View Post
    Hello this question looked straight forward enough, but I'm stuck and would really appreciate some help.

    Solve on the interval $\displaystyle 0\leq x \leq\2\pi$

    $\displaystyle \cot\theta + 3\cot2\theta-1=0$

    I tried writing it in terms of $\displaystyle \cos\theta$ and $\displaystyle \sin\theta$

    And got this;

    $\displaystyle \dfrac{5\cos^2\theat - 3\sin^2\theta}{2\sin\theta\cos\theta} =1 $, among various similar equations.

    I have worked on from here a bit, but can't seem to get it in terms of a trigonometric equation I can then solve. I'd really appreciate it if somone would tell me if I'm on the right track and should keep going or if I need to approach the question in a different way.

    Thank you very much.
    Hey Furyan!

    How about writing it in terms of $\displaystyle \cos 2\theta$ and $\displaystyle \sin 2\theta$?
    Thanks from Furyan
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  3. #3
    Member Furyan's Avatar
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    Re: cot(x) + 3cot(2x) - 1 = 0

    Hi ILikeSerena,

    Thanks, I'll try that
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  4. #4
    Member Furyan's Avatar
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    Re: cot(x) + 3cot(2x) - 1 = 0

    Hi ILikeSerena

    Thank you very much indeed, that worked! I would never have gone that way if you hadn't suggested it. Although in the end it was simple, it took me rather a long time to get there.

    I got:

    $\displaystyle 4\cos2\theta - \sin2\theta = -1$

    Then I used:

    $\displaystyle 4\cos2\theta - \sin2\theta \equiv R\cos(2\theta + \alpha)$

    And ended up with:

    $\displaystyle \sqrt{17}\cos(2\theta + \arctan\dfrac{1}{4}) = -1$

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  5. #5
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    Re: cot(x) + 3cot(2x) - 1 = 0

    cot(x) + 3cot(2x) - 1 = 0-trig-cot.png
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  6. #6
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    Re: cot(x) + 3cot(2x) - 1 = 0

    Thank you very much indeed, that worked! I would never have gone that way if you hadn't suggested it.

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