Final answer:
To solve 2sin(2x) - 2sinx + 2√3cosx - √3 = 0, we use trigonometric identities, factor the equation, and solve for x. The solution within the range of 0° to 90° is x = 60°. Thus, the only solution within the range of 0° to 90° is x = 60° which corresponds to option C.
Step-by-step explanation:
To solve the trigonometric equation 2sin(2x) - 2sinx + 2√3cosx - √3 = 0, we need to apply trigonometric identities. The double angle identity for sine is sin(2x) = 2sinx⋅cosx. Substituting this into the equation gives us:
2(2sinx⋅cosx) - 2sinx + 2√3cosx - √3 = 0
Which simplifies to:
4sinx⋅cosx - 2sinx + 2√3cosx - √3 = 0
Now, we can factor by grouping:
2sinx(2cosx - 1) + √3(2cosx - 1) = 0
(2sinx + √3)(2cosx - 1) = 0
Setting each factor equal to zero gives us two equations:
2sinx + √3 = 0 and 2cosx - 1 = 0
Solving these separately:
sinx = -√3/2, which has no solution since the sine function ranges from -1 to 1.
cosx = 1/2, which has a solution at x = 60° or x = 300° (using the unit circle).
Thus, the only solution within the range of 0° to 90° is x = 60° which corresponds to option C.