Нужно решить все номера!!

$1. a) frac{1}{2}arccos(-frac{sqrt3}{2})-4arcsin(-frac{sqrt2}{2})=frac{1}{2}(pi-arccos(frac{sqrt3}{2}))++4arcsin(frac{sqrt2}{2})=frac{1}{2}(pi-frac{pi}{6})+4frac{pi}{4}=frac{5pi}{12}+pi=frac{17pi}{12}b) tg(5arctgfrac{sqrt3}{3}-frac{1}{4}arcsinfrac{sqrt3}{2})=tg(5frac{pi}{6}-frac{1}{4}frac{pi}{3})=tg(frac{5pi}{6}-frac{pi}{12})==tg(frac{9pi}{12})=tg(frac{pi}{4})=1$
$2. a) 3sin^2 2x+2sin2x-1=0sin2x=t, -1leq tleq 13t^2+2t-1=0D=4+12=16t_1=frac{-2-4}{6}=-1 sin2x=-1 2x=frac{3pi}{2}+2pi n, nin Zx_1=frac{3pi}{4}+pi n, nin Z t_2=frac{-2+4}{6}=frac{1}{3}sin2x=frac{1}{3}2x=arcsinfrac{1}{3}+2pi k, kin Zx_2=frac{1}{2}arcsinfrac{1}{3}+pi k, kin Z$
$b) 4sin^2x+sinx *cos x-3cos^2 x=04frac{1-cos2x}{2}+frac{1}{2}sin2x-3frac{1+cos2x}{2}=04-4cos2x+sin2x-3-3cos2x=01-7cos2x+sin2x=0sqrt{1-cos^22x}=7cos2x-11-cos^22x=49cos^22x-14cos2x+150cos^22x-14cos2x=0cos2x(25cos2x-7)=0cos2x_1=0 , cos2x_2=frac{7}{25}2x_1=frac{pi}{2}+pi n, nin Z, x_1=frac{pi}{4}+frac{pi n}{2}, nin Z2x_2=arccosfrac{7}{25}+2pi k, kin Z, x_2=frac{1}{2}arccosfrac{7}{25}+pi k, kin Z$

$3. [tex]sin(frac{4x}{3}+frac{pi}{6})=-frac{1}{2}I)frac{4x}{3}+frac{pi}{6}=frac{7pi}{6}+2pi n, nin Zfrac{4x}{3}=pi +2pi n, nin Zx=frac{3pi}{4}+frac{3pi}{2}n,nin Zxin[-2pi; 2pi]-2pi leqfrac{3pi}{4}+frac{3pi}{2}n leq2pi-8leq 3+6nleq 8-11leq 6n leq 5-frac{11}{6}leq n leq frac{5}{6}n=-1; 0n=-1= textgreater x=frac{3pi}{4}-frac{3pi}{2}=-frac{3pi}{2}n=0= textgreater frac{3pi}{4}$

$II)frac{4x}{3}+frac{pi}{6}=-frac{pi}{6}+2pi k, kin Zfrac{4x}{3}=-frac{pi}{3}+2pi k, kin Zx=frac{pi}{4}+frac{3pi}{2}k, kin Zxin [-2pi; 2pi]-2pi leq frac{pi}{4}+frac{3pi}{2}kleq 2pi-8leq 1+6kleq8-9leq 6k leq 7-frac{3}{2}leq kleq frac{7}{6}k=-1; 0; 1k=-1= textgreater x=frac{pi}{4}-frac{3pi}{2}=-frac{pi}{4}k=0= textgreater x=frac{pi}{4}k=1= textgreater x=frac{7pi}{4}$

$4. left { {{sinxleq frac{sqrt3}{2}} atop {cos x textgreater -frac{1}{7}}} right. arccosfrac{1}{7}-pi textless xleq frac{pi}{3}$
Решение - графически, см. в приложении.

$5. arcsin(sqrt{x-5})=arcsin(3-sqrt{10-x})D: left { {{sqrt{x-5}leq 1} atop {-1 leq 3-sqrt{10-x}leq 1}} right. , left { {{5leq xleq 6} atop {-4leq -sqrt{10-x}leq -2}} right. , left { {{5leq xleq 6} atop {2leq sqrt{10-x}leq 4}} right. , left { {{5leq xleq 6} atop {4leq {10-x}leq 16}} right. left { {{5leq xleq 6} atop {-6leq xleq 6}} right. 5leq xleq 6 sqrt{x-5}=3-sqrt{10-x}$
$sqrt{x-5}=3-sqrt{10-x}sqrt{x-5}+sqrt{10-x}=3x-5+10-x+2sqrt{(x-5)(10-x)}=9sqrt{(x-5)(10-x)}=2(x-5)(10-x)=410x-x^2-50+5x-4=0x^2-15x+54=0x_1=7, x_2=8D: 5leq xleq 6NoSolutions$