Choose the option that represents all solutions to the given equation.
2√3(cos) sin x-4 sin2x+2√3 cos X-4=0.
1)3π x= +2 nm for every integer n. 4
2)π 4 x= +2nπ for every integer n.
3)x= +2nπ for every integer n. 2
4)3π d) x=- +2nπ for every integer n.
The correct option is:
2) x = 2nπ for every integer n.
To solve the given equation:
2√3(cos(x)) sin(x) - 4 sin^2(x) + 2√3 cos(x) - 4 = 0
Step 1: Combine like terms:
2√3(cos(x)) sin(x) - 4 sin^2(x) + 2√3 cos(x) - 4 = 0
2√3(sin(x)cos(x) + cos(x)) - 4(sin^2(x) + 1) = 0
Step 2: Use the trigonometric identity: sin^2(x) + cos^2(x) = 1
2√3(sin(x)cos(x) + cos(x)) - 4(1) = 0
2√3(sin(x)cos(x) + cos(x)) - 4 = 0
Step 3: Factor out common terms:
cos(x)[2√3sin(x) + 1] - 4 = 0
Step 4: Set each factor equal to zero and solve for x:
cos(x) = 0
x = π/2 + nπ, where n is an integer.
2√3sin(x) + 1 = 0
sin(x) = -1/(2√3)
x = -π/6 + 2nπ, where n is an integer.
So, the solutions to the given equation are:
x = π/2 + nπ, where n is an integer.
x = -π/6 + 2nπ, where n is an integer.
Therefore, the correct option is 2) x = π/4 + 2nπ for every integer n.