Question

Решите тригонометрические уравнения

1) 2cos^4x-3cos^2x+1=0

2) 4sin^4x-5sin^2x+1=0

3) 4cosx-4+sin^2x=0

4) 5sin^2x+6cosx=6

Answer 1

$1);2cos^4{x}-3cos^2{x}+1=0;cos^2{x}=aRightarrow ain begin{bmatrix}0;1end{bmatrix}\begin{Bmatrix}2a^2-3a+1=0\ain begin{bmatrix}0;1end{bmatrix}end{matrix}quad D=9-8=1quad begin{Bmatrix}a=frac{3pm 1}{4}\ain begin{bmatrix}0;1end{bmatrix}end{matrix}\\begin{Bmatrix}begin{bmatrix}a=1\a=1/2end{matrix}\ain begin{bmatrix}0;1end{bmatrix}end{matrix}quad begin{bmatrix}cos^2{x}=1\cos^2{x}=1/2end{matrix}quad begin{bmatrix}cos{x}=1\cos{x}=pm frac{sqrt{2}}{2}end{matrix}$

$begin{bmatrix}x=2pi k\x=frac{pi}{4}+frac{pi k}{2}end{matrix},kin mathbb{Z}.\\Otvet!!:;x={2pi k;frac{pi}{4}+frac{pi k}{2}},kin mathbb{Z}.$

$2);4sin^4{x}-5sin^2{x}+1=0;sin^2{x}=bRightarrow bin begin{bmatrix}0;1end{bmatrix}\begin{Bmatrix}4b^2-5b+1=0\bin begin{bmatrix}0;1end{bmatrix}end{matrix}quad D=25-16=3^2quad begin{Bmatrix}b=frac{5pm 3}{8}\bin begin{bmatrix}0;1end{bmatrix}end{matrix}\\begin{Bmatrix}begin{bmatrix}b=1\b=1/4end{matrix}\bin begin{bmatrix}0;1end{bmatrix}end{matrix}quad begin{bmatrix}sin^2{x}=1\sin^2{x}=1/4end{matrix}quad begin{bmatrix}sin{x}=1\sin{x}=pm frac{1}{2}end{matrix}$

$begin{bmatrix}x=frac{pi}{2}+2pi k\x=pm frac{pi}{6}+pi kend{matrix},kin mathbb{Z}.\\Otvet!!:;x={frac{pi}{2}+2pi k;pm frac{pi}{6}+pi k},kin mathbb{Z}.$

$3);4cos{x}-4+sin^2{x}=0\4cos{x}-4+1-cos^2{x}=0|cdot (-0,5)\0,5cos^2{x}-2cos{x}+1,5=0;D=4-3=1\cos{x}=2pm 1quad begin{Bmatrix}cos{x}=3;varnothing \cos{x}=1end{matrix}quad x=2pi k,kin mathbb{Z}.\\Otvet!!:;x=2pi k,kin mathbb{Z}.$

$4);5sin^2{x}+6cos{x}=6\5-5cos^2{x}+6cos{x}-6=0;|cdot -(0,5)\2,5cos^2{x}-3cos{x}+0,5=0;D=9-5=2^2\cos{x} =frac{3pm 2}{5}quad begin{Bmatrix}cos{x}=1\cos{x}=frac{1}{5}end{matrix}quad begin{Bmatrix}x=2pi k\x=pm arccos{frac{1}{5}}+2pi kend{matrix},kin mathbb{Z}.\\Otvet!!:;x={2pi k;pm arccos{frac{1}{5}}+2pi k},kin mathbb{Z}.$