Find general solution of the equation?

• April 30th 2009, 08:34 PM
fardeen_gen
Find general solution of the equation?
Find general solution of the equation:
$\sin \frac{2x + 1}{x} + \sin \frac{2x + 1}{3x} - 3\cos^2 \frac{2x + 1}{3x} = 0$

Spoiler:
$\frac{2}{3(2n - 1)\pi - 4}$

I am unable to solve it. Anyone?
• April 30th 2009, 11:17 PM
red_dog
Use $\sin a+\sin b=2\sin\frac{a+b}{2}\cos\frac{a-b}{2}$

$2\sin 2\left(\frac{2x+1}{3x}\right)\cos\frac{2x+1}{3x}-3\cos^2\frac{2x+1}{3x}=0$

Now use $\sin 2a=2\sin a\cos a$

$2\sin\frac{2x+1}{3x}\cos^2\frac{2x+1}{3x}-3\cos^2\frac{2x+1}{3x}=0$

$\cos^2\frac{2x+1}{3x}\left(2\sin\frac{2x+1}{3x}-3\right)=0$

Then, $\cos\frac{2x+1}{3x}=0\Rightarrow\frac{2x+1}{3x}=(2 k+1)\frac{\pi}{2}$ and find x.

If $2\sin\frac{2x+1}{3x}-3=0\Rightarrow\sin\frac{2x+1}{3x}=\frac{3}{2}>1$ and it has no solution.