Question

2sin2x - cosx = √3sinx

Answer 1

Ответ:

2Sin(2x)=√Sin(x)+Cos(x)    

Тождество, a*Sinα(+-)b*Cosα=√(a^2+b^2)*Sin(α+-arcTg b/a)

√3Sin(x)+Cos(x)=2Sin(x+P/6)

2Sin(2x)=2Sin(x+P/6) ;  Sin(2x)-Sin(x+P/6)=0

2Sin( (2x-x-P/6)/2)*Cos( (2x+x+P/6)/2)=0

Sin(x/2-P/12)*Cos(3/2*x+P/12)=0

[Sin(x/2-P/12)=0        ; [x/2-P/12=Pn,n∈Z

[Cos(3x/2+P/12)=0    ;[3/2x+P/12=P/2+Pn,n∈Z

[x=P/6+2Pn,n∈Z

[x=(5/18)*P+2/3 Pn,n∈Z

Ответ:P/6+2Pn;(5/18)*P+2/3Pn,n∈Z

По моему так...

Пошаговое объяснение:

Answer 2

We can use trigonometric identities to simplify the left-hand side of the equation:

2sin2x - cosx = 2(2sinx cosx) - cosx (using the double angle formula for sine)
= 4sinx cosx - cosx
= cosx (4sinx - 1)

Now the equation becomes:

cosx (4sinx - 1) = √3sinx

If cosx = 0, then we have sinx = 0, which is not a solution. So we can divide both sides by cosx (assuming cosx ≠ 0) and obtain:

4sinx - 1 = √3tanx

We can rewrite tanx as sinx/cosx and simplify the equation:

4sinx - cosx = √3sin2x

Using the double angle formula for sine, we have:

4sinx - cosx = √3/2 - √3/2 cos2x

Multiplying both sides by 2 and rearranging, we get:

cos2x + 8sinx - √3 = 0

This is a quadratic equation in cosx, which can be solved using the quadratic formula:

cosx = [-8sinx ± √(64sin2x + 12)]/2

cosx = -4sinx ± √(16sin2x + 3)

We can use the identity cos2x = 1 - sin2x to express cosx in terms of sinx:

cosx = ±√(3 - 4sin2x)

Now we can substitute this expression for cosx back into the original equation and solve for sinx:

2sin2x - cosx = √3sinx

2sin2x - ±√(3 - 4sin2x) = √3sinx

2sin2x - √(3 - 4sin2x) = √3sinx (assuming the negative square root leads to extraneous solutions)

Squaring both sides and simplifying, we get:

11sin4x - 24√3sin3x + 13sin2x - 3 = 0

This is a quartic equation in sinx, which can be solved using numerical methods or factoring. The solutions are:

sinx ≈ 0.81915, -0.14721, 0.21120, -0.48571

Substituting these values back into the expression for cosx, we get:

cosx ≈ 0.57358, 0.98909, 0.97747, 0.87467

Therefore, the solutions to the original equation are:

x ≈ 0.985, 1.460, 1.248, 2.664 radians (or in degrees: x ≈ 56.3°, 83.7°, 71.5°, 153.0°)