Trigonometry! #58

$3^{\sin 2x + 2\cos^{2} x}+3^{1-\sin 2x + 2\sin^{2}x}=28$

$3^{\sin 2x + 2\cos^{2} x}+3^{3-(\sin 2x + 2\cos^{2}x)}=28$

$3^{\sin 2x + 2\cos^{2}x} + \frac{3^{3}}{3^{\sin 2x + 2\cos^{2}x}}=28$

Now, let $3^{\sin 2x + 2\cos^{2} x}=a$ .

$\therefore a+\frac{27}{a}=28$

$a^{2}-28a+27=0$

$(a-27)(a-1)=0$

So either $a=27$ or $a=1$ .

For $a=27$ , we have $3^{\sin 2x + 2\cos^{2}x}=3^{3}\Rightarrow\sin 2x + 2\cos^{2} x = 3$ .

As the maximum values of the sine and cosine functions are $1$ , this equation holds true only if $\sin 2x=cos^{2}x=1$ , which is not possible. Hence this equation has no roots.

So we have $a=3$

$3^{\sin 2x + 2\cos^{2} x}=3$

$\sin 2x + 2\cos^{2}x=0$

$2\sin x\cos x + 2\cos^{2} x = 0$

$\cos x(\sin x + \cos x)=0$

$\cos x = 0 ~or~ \sin x + \cos x = 0$

For $\cos x = 0$ , $x=\frac{\pi}{2},\frac{3\pi}{2}$

For $\sin x + \cos x = 0$ ,

$\frac{1}{\sqrt{2}}\sin x + \frac{1}{\sqrt{2}}\cos x = 0$

$\sin x\cos \frac{\pi}{4} + \cos x \sin \frac{\pi}{4} = 0$

$\sin \left(\frac{\pi}{4}+x\right)=\sin 0$

$x=\frac{3\pi}{4},\frac{7\pi}{4}$

Hence, the equation has $\boxed{\text{four}}$ roots.