Final answer:
The equation √3 sin(x/2) cos(x) = 1 does not have any solutions because the sine function has a maximum value of 1, and √3 is greater than 1, making the equation impossible to satisfy within the specified range of 0° to 360°.
Step-by-step explanation:
To solve the equation √3 sin(x/2) cos(x) = 1 within the interval 0° ≤ x < 360°, we can use trigonometric identities. Firstly, recognize that the expression can be transformed using the double angle identity sin(2θ) = 2sin(θ)cos(θ). Since we have a √3 factor, we divide the double angle identity by √3:
sin(2θ) / √3 = 2sin(θ)cos(θ) / √3
This simplifies to sin(2θ) / √3 = sin(x/2) cos(x), given that θ = x/2. Now, equate this to 1:
sin(2θ) / √3 = 1
Further simplify to:
sin(2θ) = √3
2θ, or x, will be an angle where sin(2θ) = √3. But since the maximum value of the sine function is 1, there are no solutions where sin(2θ) equals √3, as √3 > 1. Hence, there are no angles x within the range 0° to 360° that satisfy the original equation.