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I got stuck on solving the following equation. I try all the identities, but hopeless. Do you have any suggestion?

$$(3x + \sin x) \cos x = -3, \quad\pi/2 \leq x \leq \pi$$ Solve for $x$.

Please don't tell me to use the calculator!

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Solutions do not appear to have a closed-form. If the only thing we can know about the zeros are approximations, then what do you want us to tell you? –  anon Apr 4 '12 at 21:13
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Why not use the calculator? There is almost certainly no "closed form" solution. So if one wants some kind of answer, one will have to use a numeical method of some kind. There is a unique solution in your interval, not far from $x=2$. –  André Nicolas Apr 4 '12 at 21:14
    
By chance, is the equation supposed to be $(3x+\sin x)\cos x=-3\pi$? Did you encounter this problem in a trig course, a calculus course, or somewhere different? –  alex.jordan Apr 4 '12 at 22:08
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1 Answer 1

I do not believe there is a closed form for this. You can only revert to numerical solutions, which are gotten from graphing the equation. One way to simplify the original is:

$$3x \cos x + \sin x \cos x = -3$$

Multiplying both sides by 2 leaves us with: $$6x \cos x + 2\sin x \cos x = -6$$

Then you could note $\sin 2x = 2 \sin x \cos x$ to get to: $$6x \cos x + \sin 2x = -6$$

From here, I would just plug in a few points and graph it to find a solution if I was told not to use a calculator.

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