# cot(x) + 3cot(2x) - 1 = 0

• Feb 15th 2013, 05:04 PM
Furyan
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 $0\leq x \leq\2\pi$

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

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

And got this;

$\dfrac{5\cos^2\theat - 3\sin^2\theta}{2\sin\theta\cos\theta} =1$, 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.
• Feb 15th 2013, 06:09 PM
ILikeSerena
Re: cot(x) + 3cot(2x) - 1 = 0
Quote:

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

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

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

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

And got this;

$\dfrac{5\cos^2\theat - 3\sin^2\theta}{2\sin\theta\cos\theta} =1$, 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 $\cos 2\theta$ and $\sin 2\theta$?
• Feb 15th 2013, 06:28 PM
Furyan
Re: cot(x) + 3cot(2x) - 1 = 0
Hi ILikeSerena,

Thanks, I'll try that:)
• Feb 15th 2013, 09:09 PM
Furyan
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:

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

Then I used:

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

And ended up with:

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

(Party)
• Feb 15th 2013, 10:26 PM
ibdutt
Re: cot(x) + 3cot(2x) - 1 = 0
• Feb 15th 2013, 10:43 PM
RolandoGamble
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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