A)
![sin3x= -\frac{ \sqrt{2} }{2} \\ 3x=(-1)^{n+1}* \frac{ \pi }{4}+ \pi n \\ x=(-1)^{n+1}* \frac{ \pi }{12}+ \frac{ \pi }{3}n](https://tex.z-dn.net/?f=sin3x%3D+-%5Cfrac%7B+%5Csqrt%7B2%7D+%7D%7B2%7D+%5C%5C+%0A3x%3D%28-1%29%5E%7Bn%2B1%7D%2A+%5Cfrac%7B+%5Cpi+%7D%7B4%7D%2B+%5Cpi+n+%5C%5C+%0Ax%3D%28-1%29%5E%7Bn%2B1%7D%2A+%5Cfrac%7B+%5Cpi+%7D%7B12%7D%2B+%5Cfrac%7B+%5Cpi+%7D%7B3%7Dn++++)
, n∈Z
б)
3cos²x-8cosx+5=0
y=cosx
3y²-8y+5=0
D=64-60=4
y₁=(8-2)/6=1
y₂=(8+2)/6=10/6=5/3=1 ²/₃
При у=1
cosx=1
x=2πn, n∈Z
При у= 1 ²/₃
cosx=1 ²/₃
Так как 1 ²/₃∉[-1; 1], то уравнение не имеет решений.
Ответ: 2πn, n∈Z.
в)
sin²x-5sinx cosx +4cos²x=0
![\frac{sin^2x}{cos^2x}- \frac{5sinxcosx}{cos^2x}+ \frac{4cos^2x}{cos^2x}= \frac{0}{cos^2x} \\ \\ tg^2x-5tgx+4=0 \\ \\ y=tgx](https://tex.z-dn.net/?f=+%5Cfrac%7Bsin%5E2x%7D%7Bcos%5E2x%7D-+%5Cfrac%7B5sinxcosx%7D%7Bcos%5E2x%7D%2B+%5Cfrac%7B4cos%5E2x%7D%7Bcos%5E2x%7D%3D+%5Cfrac%7B0%7D%7Bcos%5E2x%7D+%5C%5C+%0A+%5C%5C+%0Atg%5E2x-5tgx%2B4%3D0+%5C%5C+%0A+%5C%5C+%0Ay%3Dtgx++++)
y^2-5y+4=0
D=25-16=9
y₁=(5-3)/2=1
y₂=(5+3)/2=4
При у=1
tgx=1
x=π/4 + πn, n∈Z
При у=4
tgx=4
x=arctg4+πn, n∈Z
эти два графига надо прировнять друг к другу и решить,тогда получится точка их соединения
1) S=4πR²=36π (см²);
R²=36π/(4π)=9;
R=3 см.
Ответ: 3 см.
2)
![log_{ \sqrt{5}}25= log_{5^{ \frac{1}{2}}}25=2 log_{5}25=2*2=4. \\](https://tex.z-dn.net/?f=+log_%7B+%5Csqrt%7B5%7D%7D25%3D+log_%7B5%5E%7B+%5Cfrac%7B1%7D%7B2%7D%7D%7D25%3D2+log_%7B5%7D25%3D2%2A2%3D4.+%5C%5C++++)
Ответ: 4.
![3) 7^{3-x}\ \textgreater \ \frac{1}{49}; \\ 7^{3-x}\ \textgreater \ 7^{-2}; \\ 3-x\ \textgreater \ -2; \\ -x\ \textgreater \ -2-3; \\ -x\ \textgreater \ -5; \\ x\ \textless \ 5. \\ ](https://tex.z-dn.net/?f=3%29+7%5E%7B3-x%7D%5C+%5Ctextgreater+%5C+%5Cfrac%7B1%7D%7B49%7D%3B+%5C%5C+%0A7%5E%7B3-x%7D%5C+%5Ctextgreater+%5C+7%5E%7B-2%7D%3B+%5C%5C+%0A3-x%5C+%5Ctextgreater+%5C+-2%3B+%5C%5C+%0A-x%5C+%5Ctextgreater+%5C+-2-3%3B+%5C%5C+%0A-x%5C+%5Ctextgreater+%5C+-5%3B+%5C%5C+%0Ax%5C+%5Ctextless+%5C+5.+%5C%5C+%0A)
Ответ: (-∞;5).
![4) log^2_{2}x+ log_{2}x=0; \\ log_{2}x( log_{2}x+1)=0; \\ log_{2}x=0; \\ x=1; \\ log_{2}x+1=0; \\ log_{2}x=-1; \\ x= \frac{1}{2}. \\](https://tex.z-dn.net/?f=4%29+log%5E2_%7B2%7Dx%2B+log_%7B2%7Dx%3D0%3B+%5C%5C+%0A+log_%7B2%7Dx%28+log_%7B2%7Dx%2B1%29%3D0%3B+%5C%5C+%0A+log_%7B2%7Dx%3D0%3B+%5C%5C+%0Ax%3D1%3B+%5C%5C+%0A+log_%7B2%7Dx%2B1%3D0%3B+%5C%5C+%0A+log_%7B2%7Dx%3D-1%3B+%5C%5C+%0Ax%3D+%5Cfrac%7B1%7D%7B2%7D.+%5C%5C+++++++++)
ОДЗ:
x>0.
Ответ: 1/2; 1.
5)
![3*4^x+6^x=2*9^x; \\ ](https://tex.z-dn.net/?f=3%2A4%5Ex%2B6%5Ex%3D2%2A9%5Ex%3B+%5C%5C+%0A)
Разделить обе части уравнения на 9^x и перенесем все влево:
![3*( \frac{4}{9})^x+( \frac{6}{9} )^x-2*( \frac{9}{9} )^x=0; \\ 3*( \frac{2}{3} )^{2x}+( \frac{2}{3} )^x-2=0; \\ ( \frac{2}{3} )^x=t; \\ t\ \textgreater \ 0;\\ 3t^2+t-2=0; \\ D=1+24=25; \\ t_{1}= \frac{-1-5}{6}=-1; \\ t_{2}= \frac{-1+5}{6}= \frac{2}{3}; \\ ( \frac{2}{3} )^x= \frac{2}{3}; \\ x=1.](https://tex.z-dn.net/?f=3%2A%28+%5Cfrac%7B4%7D%7B9%7D%29%5Ex%2B%28+%5Cfrac%7B6%7D%7B9%7D+%29%5Ex-2%2A%28+%5Cfrac%7B9%7D%7B9%7D+%29%5Ex%3D0%3B+%5C%5C+%0A3%2A%28+%5Cfrac%7B2%7D%7B3%7D+%29%5E%7B2x%7D%2B%28+%5Cfrac%7B2%7D%7B3%7D+%29%5Ex-2%3D0%3B+%5C%5C+%0A%28+%5Cfrac%7B2%7D%7B3%7D+%29%5Ex%3Dt%3B++%5C%5C+%0At%5C+%5Ctextgreater+%5C+0%3B%5C%5C+%0A3t%5E2%2Bt-2%3D0%3B+%5C%5C+%0AD%3D1%2B24%3D25%3B+%5C%5C+%0A+t_%7B1%7D%3D+%5Cfrac%7B-1-5%7D%7B6%7D%3D-1%3B+%5C%5C+%0A+t_%7B2%7D%3D+%5Cfrac%7B-1%2B5%7D%7B6%7D%3D+%5Cfrac%7B2%7D%7B3%7D%3B+%5C%5C+%0A%28+%5Cfrac%7B2%7D%7B3%7D+%29%5Ex%3D+%5Cfrac%7B2%7D%7B3%7D%3B+%5C%5C+%0Ax%3D1.+++++++)
Ответ: 1.
Держи, хе :)Нужно просто раскрыть скобки и перенести в одну сторону