# Solving the equation $(3x + \sin x) \cos x = -3$

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!

-
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
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

$$3x \cos x + \sin x \cos x = -3$$
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: $$6x \cos x + \sin 2x = -6$$