Final answer:
To solve the equation √3 sin(x/2) cos(x) = 1, simplify the equation using trigonometric identities and find the values of x that satisfy it within the given range. The solutions are x = 60° and x = 120°.
Step-by-step explanation:
To solve the equation √3 sin(x/2) cos(x) = 1, we need to find the values of x that satisfy this equation within the given range 0° ≤ x < 360°.
First, let's simplify the equation. We know that √3 is a constant, so we can divide both sides of the equation by √3 to get sin(x/2) cos(x) = 1/√3.
Next, we can use trigonometric identities to simplify further. We know that sin(2a) = 2sin(a)cos(a), so sin(x/2) cos(x) can be written as sin(x/2) cos(x) = sin(x)cos(x/2).
Substituting this into the equation, we have sin(x)cos(x/2) = 1/√3.
Now, let's solve for x by considering the possible values of x within the given range:
- When sin(x) = 1/√3 and cos(x/2) = 1, x = 60°.
- When sin(x) = -1/√3 and cos(x/2) = -1, x = 120°.
Therefore, the solutions to the equation within the given range are x = 60° and x = 120°. So options 2 (120°) and 3 (60°) are correct.