1)
2sin(x/2)=3sin²(x/2)
2sin(x/2)-3sin²(x/2)=0
sin(x/2) (2-3sin(x/2))=0
a) sin(x/2)=0
x/2=πk, k∈Z
x=2πk, k∈Z
b) 2-3sin(x/2)=0
-3sin(x/2)=-2
sin(x/2)=2/3
x/2=(-1)^n * arcsin(2/3)+πk, k∈Z
x=2*(-1)^n * arcsin(2/3)+2πk, k∈Z
Ответ: 2πk, k∈Z;
2*(-1)^k*arcsin(2/3)+2πk, k∈Z.
2)
sin6xcosx+cos6xsinx=0.5
sin(6x+x)=0.5
sin7x=0.5
7x=(-1)^k*(π/6)+πk, k∈Z
x=(-1)^k*(π/42)+(π/7)*k, k∈Z
Ответ: (-1)^k*(π/42)+(π/7)*k, k∈Z.
3)
3sinx+4sin(π/2+x)=0
3sinx+4cosx=0
![3sin2*( \frac{x}{2} )+4cos2*( \frac{x}{2} )=0 \\ \\ 3*2sin( \frac{x}{2} )cos( \frac{x}{2} )+4(cos^2( \frac{x}{2} )-sin^2( \frac{x}{2} ))=0 \\ \\ -4sin^2( \frac{x}{2} )+6sin( \frac{x}{2} )cos( \frac{x}{2} )+4cos^2( \frac{x}{2} )=0 \\ \\ 2sin^2( \frac{x}{2} )-3sin( \frac{x}{2} )cos( \frac{x}{2} )+2cos^2( \frac{x}{2} )=0 \\ \\ \frac{2sin^2( \frac{x}{2} )}{cos^2( \frac{x}{2} )}- \frac{3sin( \frac{x}{2} )cos( \frac{x}{2} )}{cos^2( \frac{x}{2} )}+ \frac{2cos^2( \frac{x}{2} )}{cos^2( \frac{x}{2} )}](https://tex.z-dn.net/?f=3sin2%2A%28+%5Cfrac%7Bx%7D%7B2%7D+%29%2B4cos2%2A%28+%5Cfrac%7Bx%7D%7B2%7D+%29%3D0+%5C%5C++%5C%5C+%0A3%2A2sin%28+%5Cfrac%7Bx%7D%7B2%7D+%29cos%28+%5Cfrac%7Bx%7D%7B2%7D+%29%2B4%28cos%5E2%28+%5Cfrac%7Bx%7D%7B2%7D+%29-sin%5E2%28+%5Cfrac%7Bx%7D%7B2%7D+%29%29%3D0+%5C%5C++%5C%5C+%0A-4sin%5E2%28+%5Cfrac%7Bx%7D%7B2%7D+%29%2B6sin%28+%5Cfrac%7Bx%7D%7B2%7D+%29cos%28+%5Cfrac%7Bx%7D%7B2%7D+%29%2B4cos%5E2%28+%5Cfrac%7Bx%7D%7B2%7D+%29%3D0+%5C%5C++%5C%5C+%0A2sin%5E2%28+%5Cfrac%7Bx%7D%7B2%7D+%29-3sin%28+%5Cfrac%7Bx%7D%7B2%7D+%29cos%28+%5Cfrac%7Bx%7D%7B2%7D+%29%2B2cos%5E2%28+%5Cfrac%7Bx%7D%7B2%7D+%29%3D0+%5C%5C++%5C%5C+%0A+%5Cfrac%7B2sin%5E2%28+%5Cfrac%7Bx%7D%7B2%7D+%29%7D%7Bcos%5E2%28+%5Cfrac%7Bx%7D%7B2%7D+%29%7D-+%5Cfrac%7B3sin%28+%5Cfrac%7Bx%7D%7B2%7D+%29cos%28+%5Cfrac%7Bx%7D%7B2%7D+%29%7D%7Bcos%5E2%28+%5Cfrac%7Bx%7D%7B2%7D+%29%7D%2B+%5Cfrac%7B2cos%5E2%28+%5Cfrac%7Bx%7D%7B2%7D+%29%7D%7Bcos%5E2%28+%5Cfrac%7Bx%7D%7B2%7D+%29%7D+++)
=0
![2tg^2( \frac{x}{2} )-3tg( \frac{x}{2} )-2=0 \\ \\ y=tg( \frac{x}{2} ) \\ \\ 2y^2-3y-2=0 \\ D=9+4*2*2=25 \\ y_{1} =\frac{3-5}{4}=- \frac{2}{4}=- \frac{1}{2} \\ \\ y_{2}= \frac{3+5}{4}=2](https://tex.z-dn.net/?f=2tg%5E2%28+%5Cfrac%7Bx%7D%7B2%7D+%29-3tg%28+%5Cfrac%7Bx%7D%7B2%7D+%29-2%3D0+%5C%5C++%5C%5C+%0Ay%3Dtg%28+%5Cfrac%7Bx%7D%7B2%7D+%29+%5C%5C++%5C%5C+%0A2y%5E2-3y-2%3D0+%5C%5C+%0AD%3D9%2B4%2A2%2A2%3D25+%5C%5C+%0Ay_%7B1%7D+%3D%5Cfrac%7B3-5%7D%7B4%7D%3D-+%5Cfrac%7B2%7D%7B4%7D%3D-+%5Cfrac%7B1%7D%7B2%7D+%5C%5C++%5C%5C+%0Ay_%7B2%7D%3D+%5Cfrac%7B3%2B5%7D%7B4%7D%3D2++++)
a) При у=-1/2
![tg( \frac{x}{2} )=- \frac{1}{2} \\ \frac{x}{2}=-arctg \frac{1}{2} + \pi k \\ \\ x=-2arctg \frac{1}{2}+2 \pi k](https://tex.z-dn.net/?f=tg%28+%5Cfrac%7Bx%7D%7B2%7D+%29%3D-+%5Cfrac%7B1%7D%7B2%7D+%5C%5C+%0A+%5Cfrac%7Bx%7D%7B2%7D%3D-arctg+%5Cfrac%7B1%7D%7B2%7D+%2B+%5Cpi+k+%5C%5C++%5C%5C+%0Ax%3D-2arctg+%5Cfrac%7B1%7D%7B2%7D%2B2+%5Cpi+k+++)
,
k∈Z;
b) При у=2
![tg( \frac{x}{2} )=2 \\ \frac{x}{2} =arctg2+ \pi k \\ \\ x=2arctg2+2 \pi k,](https://tex.z-dn.net/?f=tg%28+%5Cfrac%7Bx%7D%7B2%7D+%29%3D2+%5C%5C+%0A+%5Cfrac%7Bx%7D%7B2%7D+%3Darctg2%2B+%5Cpi+k+%5C%5C++%5C%5C+%0Ax%3D2arctg2%2B2+%5Cpi+k%2C)
k∈Z.
Ответ:
![-2arctg \frac{1}{2}+2 \pi k,](https://tex.z-dn.net/?f=-2arctg+%5Cfrac%7B1%7D%7B2%7D%2B2+%5Cpi+k%2C++)
k∈Z;
![2arctg2+2 \pi k,](https://tex.z-dn.net/?f=2arctg2%2B2+%5Cpi+k%2C)
k∈Z.
<span>y=x-5.
x 0 5
--------------
y -5 0
Через точки (0; -5) и (5;0) строишь прямую.
</span>
a) наибольшее У=-2 и наименьшее значение У = -5 функции на отрезке [0;3];
б) значение переменной x, при которых y=0 х=5; y>0 х>5
надо подобрать такие числа, чтоб в разности 5х-3у получилось -18 или 18
5*3-3*11+18=0
5*9-3*9=0
или выразить одну переменную, через другую
Tg*ctg=1
подставляем:
(-3/5)*(-5/8)=1
3/8=1
неверно, значит не могут