Question

1)6cos^2x-cosx-2/корень из -sin=02)корень из cos2x=1+2sinx3)sinx+sin2x/sin3x=-1помогите решить

Answer 1

1)sinx<0⇒x∈(π+2πn;2π+2πn)
6cos²x-cosx-2=0
cosx=a
6a²-a-2=0
D=1+48=49
a1=(1-7)/12=-1/2⇒cosx=-1/2⇒x=-2π/3+2πn U x=2π/3+2πn
x=4π/3+2πn∈(π+2πn;2π+2πn)
a2=(1+7)/12=2/3⇒cosx=-arccos2/3+2πn U x=arccoax+2πn
x=3/2+-accos2/3+2πn∈(π+2πn;2π+2πn)
2)1+2sinx≥0⇒sinx≥-1/2⇒x∈[-π/3+2πn;4π/3+2πn]
cos2x=1+4sinx+4sin²x
1-2sin²x=1+4sinx+4sin²x
6sin²x+4sinx=0
2sinx(3sinx+2)=0
sinx=0⇒x=πn
sinx=-2/3 x=(-1)^n+1*arcsinx+πn∉[-π/3+2πn;4π/3+2πn]
3)sin3x≠0⇒x≠πn/3
2sin(3x/2)cos(x/2)/2sin(3x/2)cos(3x/2)=-1
cos(x/2)/cos(3x/2)=-1
cos(x/2)+cos(3x/2)=0
2cosxcos(x/2)=0
cosx=0⇒x=π/2+πn
cosx/2=0⇒x/2=π/2+πn⇒x=π+2πn

Answer 2

1) To solve the equation 6cos^2x - cosx - 2/√(-sin) = 0, we can follow these steps:

Step 1: Simplify the equation by multiplying both sides by √(-sin) to get rid of the denominator: 6cos^2x - cosx - 2 = 0.

Step 2: Rearrange the equation to form a quadratic equation in terms of cosx: 6cos^2x - cosx - 2 = 0.

Step 3: Solve the quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, factoring seems to be the most straightforward approach.

The factored form of the equation is: (2cosx + 1)(3cosx - 2) = 0.

Setting each factor equal to zero gives us two possible solutions: 2cosx + 1 = 0 or 3cosx - 2 = 0.

Solving these equations individually will give us the values of x that satisfy the original equation.

For 2cosx + 1 = 0: 2cosx = -1 cosx = -1/2 x = π/3 + 2πn or x = 5π/3 + 2πn, where n is an integer.

For 3cosx - 2 = 0: 3cosx = 2 cosx = 2/3 x = arccos(2/3) + 2πn or x = -arccos(2/3) + 2πn, where n is an integer.

2) To solve the equation √(cos2x) = 1 + 2sinx, we can follow these steps:

Step 1: Square both sides of the equation to eliminate the square root: cos2x = (1 + 2sinx)^2.

Step 2: Expand the right side of the equation: cos2x = 1 + 4sinx + 4sin^2x.

Step 3: Use the identity cos2x = 2cos^2x - 1 to rewrite the left side of the equation: 2cos^2x - 1 = 1 + 4sinx + 4sin^2x.

Step 4: Rearrange the equation to form a quadratic equation in terms of sinx: 4sin^2x + 4sinx - 2cos^2x = 0.

Step 5: Use the identity sin^2x + cos^2x = 1 to simplify the equation: 4sin^2x + 4sinx - 2(1 - sin^2x) = 0.

Step 6: Simplify further: 6sin^2x + 4sinx - 2 = 0.

Step 7: Solve the quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, factoring seems to be the most straightforward approach.

The factored form of the equation is: (3sinx - 1)(2sinx + 2) = 0.

Setting each factor equal to zero gives us two possible solutions: 3sinx - 1 = 0 or 2sinx + 2 = 0.

Solving these equations individually will give us the values of x that satisfy the original equation.

For 3sinx - 1 = 0: 3sinx = 1 sinx = 1/3 x = arcsin(1/3) + 2πn or x = π - arcsin(1/3) + 2πn, where n is an integer.

For 2sinx + 2 = 0: 2sinx = -2 sinx = -1 x = arcsin(-1) + 2πn or x = π - arcsin(-1) + 2πn, where n is an integer.

3) To solve the equation sinx + sin2x/sin3x = -1, we can follow these steps:

Step 1: Multiply both sides of the equation by sin3x to eliminate the denominator: sinx * sin3x + sin2x = -sin3x.

Step 2: Expand the left side of the equation: sinx * (3sinx - 4sin^3x) + sin2x = -sin3x.

Step 3: Simplify the equation: 3sin^2x - 4sin^4x + sin2x = -sin3x.

Step 4: Use the identity sin2x = 2sinx*cosx to rewrite the equation: 3sin^2x - 4sin^4x + 2sinx*cosx = -sin3x.

Step 5: Rearrange the equation to form a quadratic equation in terms of sinx: 4sin^4x - 3sin^2x + sin3x + 2sinx*cosx = 0.

Step 6: Solve the quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, factoring seems to be the most straightforward approach.

The factored form of the equation is: sinx(4sin^3x - 3sinx + cosx) + sin3x = 0.

Setting each factor equal to zero gives us two possible solutions: sinx = 0 or 4sin^3x - 3sinx + cosx = 0.

For sinx = 0: x = 0 + πn or x = π + πn, where n is an integer.

For 4sin^3x - 3sinx + cosx = 0, finding an exact solution may be challenging. Numerical methods or approximations may be needed to find the values of x that satisfy this equation.

Please note that these solutions are based on the given equations and may not cover all possible solutions.