Каким бы ни было значение А, это выражение в любом случае имеет значение -1 :
раскроем скобки:
а^3+a^2+a-a^2-a-1-a^3
теперь сократим одинаковые члены с разным знаком (+) и (-) например: а^2 и -a^2:
<u>а^3</u>+<u>a^2</u>+<u>a</u>-<u>a^2</u>-<u>a</u>-1-<u>a^3</u> (подчеркнутое вычеркиваем из выражения)
остается:
-1
таким образом, значение А на выражение не влияет.
![\sin(2x- \frac{7\pi}{2})+\sin( \frac{3 \pi }{2} -8x)+\cos6x=1\\ \cos 2x-\cos8x+\cos6x=1\\ \cos2x-2\cos^24x+1+2\cos^23x-1=1\\ 2\cos^2x-1-2((2\cos^22x-1)^2)^2+2(4\cos^3x-3\cos x)^2=1\\ 2\cos^2x-1-2((2(2\cos^2x-1)^2-1)^2+2(4\cos^3x-3\cos x)^2=1\\](https://tex.z-dn.net/?f=%5Csin%282x-+%5Cfrac%7B7%5Cpi%7D%7B2%7D%29%2B%5Csin%28+%5Cfrac%7B3+%5Cpi+%7D%7B2%7D++-8x%29%2B%5Ccos6x%3D1%5C%5C+%5Ccos+2x-%5Ccos8x%2B%5Ccos6x%3D1%5C%5C+%5Ccos2x-2%5Ccos%5E24x%2B1%2B2%5Ccos%5E23x-1%3D1%5C%5C+2%5Ccos%5E2x-1-2%28%282%5Ccos%5E22x-1%29%5E2%29%5E2%2B2%284%5Ccos%5E3x-3%5Ccos+x%29%5E2%3D1%5C%5C+2%5Ccos%5E2x-1-2%28%282%282%5Ccos%5E2x-1%29%5E2-1%29%5E2%2B2%284%5Ccos%5E3x-3%5Ccos+x%29%5E2%3D1%5C%5C+)
Пусть
![\cos x = t\,(|t| \leq 1)](https://tex.z-dn.net/?f=%5Ccos+x+%3D+t%5C%2C%28%7Ct%7C+%5Cleq+1%29)
, тогда получаем
![2t^2-2(2(2t^2-1)^2-1)^2+2(4t^3-3t)^2-1=1](https://tex.z-dn.net/?f=2t%5E2-2%282%282t%5E2-1%29%5E2-1%29%5E2%2B2%284t%5E3-3t%29%5E2-1%3D1)
![2t^2-2(8t^4-8t^2+1)^2+2(4t^3-3t^2)-2=0\\ 2t^2-2(64t^8-128t^6+80t^4-16t^2+1)+2(16t^6-24t^4+9t^2)-2=0\\ 2t^2-128t^8+256t^6-160t^4+32t^2-2+32t^6-48t^4+18t^2-2=0\\128t^8-288t^2+208t^4-52t^2+4=0|:4\\ 32t^8-72t^6+52t^4-13t^2+1=0](https://tex.z-dn.net/?f=2t%5E2-2%288t%5E4-8t%5E2%2B1%29%5E2%2B2%284t%5E3-3t%5E2%29-2%3D0%5C%5C+2t%5E2-2%2864t%5E8-128t%5E6%2B80t%5E4-16t%5E2%2B1%29%2B2%2816t%5E6-24t%5E4%2B9t%5E2%29-2%3D0%5C%5C+2t%5E2-128t%5E8%2B256t%5E6-160t%5E4%2B32t%5E2-2%2B32t%5E6-48t%5E4%2B18t%5E2-2%3D0%5C%5C128t%5E8-288t%5E2%2B208t%5E4-52t%5E2%2B4%3D0%7C%3A4%5C%5C+32t%5E8-72t%5E6%2B52t%5E4-13t%5E2%2B1%3D0)
Пусть
![t^2=z\,(z \geq 0)](https://tex.z-dn.net/?f=t%5E2%3Dz%5C%2C%28z+%5Cgeq+0%29)
, тогда получаем
![32z^4-72z^3+52z^2-13z+1=0\\ (32z^4-72z^3+1)+(52z^2-13z)=0\\ (4z-1)(8z^3-16z^2-4z-1)+13z(4z-1)=0\\ (4z-1)(8z^3-16z^2+9z-1)=0\\ z_1= \frac{1}{4} \\ 8z^3-16z^2+9z-1=0](https://tex.z-dn.net/?f=32z%5E4-72z%5E3%2B52z%5E2-13z%2B1%3D0%5C%5C+%2832z%5E4-72z%5E3%2B1%29%2B%2852z%5E2-13z%29%3D0%5C%5C+%284z-1%29%288z%5E3-16z%5E2-4z-1%29%2B13z%284z-1%29%3D0%5C%5C+%284z-1%29%288z%5E3-16z%5E2%2B9z-1%29%3D0%5C%5C+z_1%3D+%5Cfrac%7B1%7D%7B4%7D+%5C%5C+8z%5E3-16z%5E2%2B9z-1%3D0)
Разложим на множители
![8z^3-8z^2-8z^2+8z+z-1=0\\ 8z^2(z-1)-8z(z-1)+(z-1)=0\\ (z-1)(8z^2-8z+1)=0\\ z_2=1\\ 8z^2-8z+1=0](https://tex.z-dn.net/?f=8z%5E3-8z%5E2-8z%5E2%2B8z%2Bz-1%3D0%5C%5C+8z%5E2%28z-1%29-8z%28z-1%29%2B%28z-1%29%3D0%5C%5C+%28z-1%29%288z%5E2-8z%2B1%29%3D0%5C%5C+z_2%3D1%5C%5C+8z%5E2-8z%2B1%3D0)
Находим дискриминант
![D=b^2-4ac=(-8)^2-4\cdot8\cdot1=32;\,\, \sqrt{D} =4 \sqrt{2} \\ z_3_,_4= \frac{2- \sqrt{2} }{4}](https://tex.z-dn.net/?f=D%3Db%5E2-4ac%3D%28-8%29%5E2-4%5Ccdot8%5Ccdot1%3D32%3B%5C%2C%5C%2C+%5Csqrt%7BD%7D+%3D4+%5Csqrt%7B2%7D+%5C%5C+z_3_%2C_4%3D+%5Cfrac%7B2-+%5Csqrt%7B2%7D+%7D%7B4%7D+)
Возвращаемся к замене от z
![t^2=\frac{1}{4}\\ t=\pm\frac{1}{2}\\ \\ t^2=1\\ t=\pm1\\ \\ t^2= \frac{2- \sqrt{2} }{4}\\ t=\pm \frac{\sqrt{2- \sqrt{2}} }{2}](https://tex.z-dn.net/?f=t%5E2%3D%5Cfrac%7B1%7D%7B4%7D%5C%5C+t%3D%5Cpm%5Cfrac%7B1%7D%7B2%7D%5C%5C+%5C%5C+t%5E2%3D1%5C%5C+t%3D%5Cpm1%5C%5C+%5C%5C+t%5E2%3D+%5Cfrac%7B2-+%5Csqrt%7B2%7D+%7D%7B4%7D%5C%5C+t%3D%5Cpm+%5Cfrac%7B%5Csqrt%7B2-+%5Csqrt%7B2%7D%7D+%7D%7B2%7D)
Возвращаемся к замене от t
![\cos x=1\\ x=2 \pi n,n \in Z\\\\ \cos x=-1\\ x= \pi +2 \pi n,n \in Z\\\\ \cos x=-0.5\\ x=\pm \frac{2 \pi }{3} +2 \pi n,n \in Z\\ \\ \cos x=0.5\\ x=\pm \frac{\pi}{3}+2 \pi n,n \in Z \\ \\ \cos x = \frac{\sqrt{2- \sqrt{2}} }{2}\\ x=\pm \arccos(\frac{\sqrt{2- \sqrt{2}} }{2})+2 \pi n,n \in Z](https://tex.z-dn.net/?f=%5Ccos+x%3D1%5C%5C+x%3D2+%5Cpi+n%2Cn+%5Cin+Z%5C%5C%5C%5C+%5Ccos+x%3D-1%5C%5C+x%3D+%5Cpi+%2B2+%5Cpi+n%2Cn+%5Cin+Z%5C%5C%5C%5C++%5Ccos+x%3D-0.5%5C%5C+x%3D%5Cpm+%5Cfrac%7B2+%5Cpi+%7D%7B3%7D+%2B2+%5Cpi+n%2Cn+%5Cin+Z%5C%5C+%5C%5C+%5Ccos+x%3D0.5%5C%5C+x%3D%5Cpm+%5Cfrac%7B%5Cpi%7D%7B3%7D%2B2+%5Cpi+n%2Cn+%5Cin+Z+%5C%5C+%5C%5C+%5Ccos+x+%3D+%5Cfrac%7B%5Csqrt%7B2-+%5Csqrt%7B2%7D%7D+%7D%7B2%7D%5C%5C+x%3D%5Cpm+%5Carccos%28%5Cfrac%7B%5Csqrt%7B2-+%5Csqrt%7B2%7D%7D+%7D%7B2%7D%29%2B2+%5Cpi+n%2Cn+%5Cin+Z)
![\\ \cos x=-\frac{\sqrt{2- \sqrt{2}} }{2}\\ x=\pm \arccos(-\frac{\sqrt{2- \sqrt{2}} }{2})+2 \pi n,n \in Z](https://tex.z-dn.net/?f=%5C%5C+%5Ccos+x%3D-%5Cfrac%7B%5Csqrt%7B2-+%5Csqrt%7B2%7D%7D+%7D%7B2%7D%5C%5C+x%3D%5Cpm+%5Carccos%28-%5Cfrac%7B%5Csqrt%7B2-+%5Csqrt%7B2%7D%7D+%7D%7B2%7D%29%2B2+%5Cpi+n%2Cn+%5Cin+Z)
-3x+26≥23
-3x≥23-26
-3x≥-3
x≤1
x∈(-∞;1]