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Title says it all: $$2\cos(\theta)+(\theta)=0$$ the interval should be between $0$ to $2\pi$.

Been trying to figure this out for quite a while, still no luck. I'm trying to find if the solution exists or not.

Michael Hardy
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Kapooky Handy
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  • I suppose numerical methods are out of the question? – Oria Gruber Sep 28 '14 at 20:03
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    Title doesn't say it all; do you want to find such an $x$ or just show that such an $x$ exists? – Quinn Culver Sep 28 '14 at 20:03
  • I'm inclined to believe there is no analytical way to find such an $x$. there are however a plethora of numerical methods to solve questions like this. – Oria Gruber Sep 28 '14 at 20:05
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    The title definitely doesn't say what you want. If the problem is to solve for $x$, you should say so. If the problem is to figure out how many solutions exist, you should say so. If the problem is to prove that the solution is an irrational number, you should say so. If the problem is to ask whether there is a closed form for the solution, you should say so. If the problem is which numerical method to use, you should say so. – Michael Hardy Sep 28 '14 at 20:06
  • A numerical method is de rigeur. – ncmathsadist Sep 28 '14 at 20:06
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    You can only find a solution numerically http://www.wolframalpha.com/input/?i=2cosx%2Bx%3D0 – Stefano Sep 28 '14 at 20:07
  • Sorry about the question not being specific, I edited it for better understanding. I'm trying to find the solutions(In radians) that exist. We are supposed to use a few advanced trig identities to help us out. – Kapooky Handy Sep 28 '14 at 20:13
  • @KapookyHandy Understand that an equation is not a question. There is no question in your question. – Andrew Dudzik Sep 28 '14 at 20:16
  • I don't quite understand what you mean. If I was to say ''find the solutions of 2cos(pheta)+(pheta)=0'' would that make more sense? – Kapooky Handy Sep 28 '14 at 20:20
  • That would make more sense. – Michael Hardy Sep 28 '14 at 20:30

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You say the title says it all, but that is entirely wrong, as I explained in a comment.

I will guess that the problem is either to find a solution or to show that one exists.

CAREFULLY draw the graph of $y=2\cos x$. Then CAREFULLY (this one's easy) draw the graph of $y=-x$.

That will tell you that exactly one solution exists, and it will tell you approximately what number $x$ is.

My guess is Newton's method will converge quickly.

Michael Hardy
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  • Out of curiosity, how hard is it to prove (or disprove) that the solution is irrational? (perhaps I'll post a question) – Sheheryar Zaidi Sep 28 '14 at 20:14
  • @SheheryarZaidi You want to show that $\cos(t)$ is irrational when $t\neq 0$ is rational. This can probably be done by looking at the power series for cosine, and imitating the standard proof that $e$ is irrational. – Andrew Dudzik Sep 28 '14 at 20:18
  • @Slade : I suspect a proof of this is in a certain book by Ivan Niven that I have at hand. – Michael Hardy Sep 28 '14 at 20:20
  • @Slade: Why do we want to show that? Let's assume that a rational argument produces an irrational output for cosine, then the solution satisfies $\cos x = -x/2$, if $x$ is rational, then $\cos x$ is irrational but $-x/2$ is rational. Maybe I'm missing something? – Sheheryar Zaidi Sep 28 '14 at 20:26
  • @Slade: Ahh, it makes perfect sense now! We get a contradiction, nice. – Sheheryar Zaidi Sep 28 '14 at 20:27