Solve the equation $4 \sin x \cos (2x) \sin (3x) = 1$ for $0^{\circ} < x < 180^{\circ}$

Question

I have tried using identities but none of them worked for me.

Eg: $4 \sin x(1 - 2 \sin^2 x)(3 \sin x - 4 \sin^3 x) = 1$

Answer 1

If $\cos{x}=0$ so $x=90^{\circ},$ which is not a root of the equation.

Thus, for $x\neq90^{\circ}$ our equation is equivalent to $$4\cos{x}\sin{x}\cos2x\sin3x=\cos{x}$$ or $$\sin4x\sin3x=\cos{x}$$ or $$\cos{x}-\cos7x=2\cos{x}$$ or $$\cos7x+\cos{x}=0.$$ Can you end it now?

Answer 2

Hint: It is $$4\sin(x)\cos(2x)\sin(3x)-1=$$$$-2 \sin \left(\frac{\pi }{4}-2 x\right) \sin \left(2 x+\frac{\pi }{4}\right) (2 \cos (2 x)-1)$$

Answer 3

$$4 \sin(x)\cos(2x)\sin(3x) - 1 = 32 \sin(x)^6 - 40 \sin(x)^4 + 12 \sin(x)^2 - 1$$ Letting $\sin(x) = s$, this is $$ 32 s^6 - 40 s^4 + 12 s^2 - 1 = (2s+1)(2s-1)(8s^4 - 8s^2+1)$$ which has roots $$ s = \pm \frac{1}{2},\; \pm \frac{\sqrt{2\pm\sqrt{2}}}{2} $$ The values of $x$ in $(0,\pi)$ are $$\frac{\pi}{8},\; \frac{\pi}{6},\; \frac{3\pi}{8},\; \frac{5\pi}{8}\; \frac{5\pi}{6},\; \frac{7\pi}{8}$$ or in degrees, $$22.5,\; 30,\; 67.5,\; 112.5,\; 150,\; 157.5 $$

Answer 4

Hint:

Make a partial linearisation: $\;2\sin x\sin 3x=\cos(3x-x)-\cos(3x+x)$, so \begin{align} 4 \sin x \cos (2x) \sin (3x)&=2\cos 2x(\cos 2x-\cos 4x)=2\cos^22x-2\cos 2x\cos 4x\\ &=1+\cos 4x-2\cos 2x\cos 4x, \end{align} and after simplification, the equation becomes $$\cos 4x(1-2\cos 2x)=0.$$ Can you solve this one?