since sin2x+√3cos2x=2sin(2x+60) find the general solution for the equation sin2x+√3cosx=0
thank you
since sin(n*180) = 0,
2x+60 = n*180
x = n*90 - 30
To solve the equation sin2x + √3cosx = 0, we can start by rewriting sin2x as 2sinxcosx:
2sinxcosx + √3cosx = 0
Now, let's factor out the common term cosx:
cosx (2sinx + √3) = 0
To find the solutions, we set each factor equal to zero and solve for x:
1) cosx = 0:
Using the unit circle, we know that cosx = 0 when x = ±nπ/2, where n is an integer.
2) 2sinx + √3 = 0:
Subtracting √3 from both sides:
2sinx = -√3
Dividing both sides by 2:
sinx = -√3/2
Using the unit circle or reference angles, we find that sinx = -√3/2 when x = -π/3 + 2nπ or x = -2π/3 + 2nπ, where n is an integer.
Therefore, the general solution for the equation sin2x + √3cosx = 0 is:
x = ±nπ/2 or x = -π/3 + 2nπ or x = -2π/3 + 2nπ, where n is an integer.