\sqrt2\\\\\star \ \ 2\, sina\cdot cosa=sin2a\ \ \to \ \ \ sina\cdot cosa=\dfrac{1}{2}\, sin2a\ \ \star \\\\2\cdot sin6x>\sqrt2\\\\sin\, 6x>\dfrac{\sqrt2}{2}\ \ \ \Rightarrow \ \ \ \dfrac{\pi}{4}+2\pi n" alt="1)\ \ 4\, sin3x\cdot cos3x>\sqrt2\\\\\star \ \ 2\, sina\cdot cosa=sin2a\ \ \to \ \ \ sina\cdot cosa=\dfrac{1}{2}\, sin2a\ \ \star \\\\2\cdot sin6x>\sqrt2\\\\sin\, 6x>\dfrac{\sqrt2}{2}\ \ \ \Rightarrow \ \ \ \dfrac{\pi}{4}+2\pi n" align="absmiddle" class="latex-formula">
![3)\ \ 3sin2x+8cos^2x\geq 7\\\\6\cdot sinx\cdot cosx+8cos^2x-7(sin^2x+cos^2x)\geq 0\\\\-7sin^2x+6\cdot sinx\cdot cosx+cos^2x\geq 0\ \Big|:cos^2x\ne 0\\\\-7tg^2x+6tgx+1\geq 0\ \ \ \Rightarrow \ \ \ 7tg^2x-6tgx-1\leq 0\ ,\\\\t=tgx\ \ ,\ \ \ 7t^2-6t-1\leq 0\ \ ,\ \ t_1=-\dfrac{1}{7}\ \ ,\ \ t_2=1\ ,\\\\7(t+\dfrac{1}{7})(t-1)\leq 0\ \ ,\ \ \ \ +++[-\dfrac{1}{7}\ ]---[\ 1\ ]+++](https://tex.z-dn.net/?f=3%29%5C%20%5C%203sin2x%2B8cos%5E2x%5Cgeq%207%5C%5C%5C%5C6%5Ccdot%20sinx%5Ccdot%20cosx%2B8cos%5E2x-7%28sin%5E2x%2Bcos%5E2x%29%5Cgeq%200%5C%5C%5C%5C-7sin%5E2x%2B6%5Ccdot%20sinx%5Ccdot%20cosx%2Bcos%5E2x%5Cgeq%200%5C%20%5CBig%7C%3Acos%5E2x%5Cne%200%5C%5C%5C%5C-7tg%5E2x%2B6tgx%2B1%5Cgeq%200%5C%20%5C%20%5C%20%5CRightarrow%20%5C%20%5C%20%5C%207tg%5E2x-6tgx-1%5Cleq%200%5C%20%2C%5C%5C%5C%5Ct%3Dtgx%5C%20%5C%20%2C%5C%20%5C%20%5C%207t%5E2-6t-1%5Cleq%200%5C%20%5C%20%2C%5C%20%5C%20t_1%3D-%5Cdfrac%7B1%7D%7B7%7D%5C%20%5C%20%2C%5C%20%5C%20t_2%3D1%5C%20%2C%5C%5C%5C%5C7%28t%2B%5Cdfrac%7B1%7D%7B7%7D%29%28t-1%29%5Cleq%200%5C%20%5C%20%2C%5C%20%5C%20%5C%20%5C%20%2B%2B%2B%5B-%5Cdfrac%7B1%7D%7B7%7D%5C%20%5D---%5B%5C%201%5C%20%5D%2B%2B%2B "3)\ \ 3sin2x+8cos^2x\geq 7\\\\6\cdot sinx\cdot cosx+8cos^2x-7(sin^2x+cos^2x)\geq 0\\\\-7sin^2x+6\cdot sinx\cdot cosx+cos^2x\geq 0\ \Big|:cos^2x\ne 0\\\\-7tg^2x+6tgx+1\geq 0\ \ \ \Rightarrow \ \ \ 7tg^2x-6tgx-1\leq 0\ ,\\\\t=tgx\ \ ,\ \ \ 7t^2-6t-1\leq 0\ \ ,\ \ t_1=-\dfrac{1}{7}\ \ ,\ \ t_2=1\ ,\\\\7(t+\dfrac{1}{7})(t-1)\leq 0\ \ ,\ \ \ \ +++[-\dfrac{1}{7}\ ]---[\ 1\ ]+++")
