To find the solution to the equation (2cos x - √3)(2sin x +1) = 0, we use the zero product property to solve for x where either 2cos x - √3 = 0 or 2sin x + 1 = 0. Solutions for each part come from familiar values on the unit circle.
To solve the equation (2cos x - √3)(2sin x +1) = 0, we need to apply the zero product property, which states that if a product of two factors is zero, then at least one of the factors must be zero. Therefore, we can set each factor equal to zero and solve the equations separately:
2cos x - √3 = 0
2sin x + 1 = 0
We first address 2cos x - √3 = 0. Adding √3 to both sides gives us 2cos x = √3, and dividing both sides by 2 gives us cos x = √3/2. From the unit circle, we know that cos x = √3/2 for x = 30° or x = 330° (in degrees) or x = π/6 or x = 11π/6 (in radians).
Next, for 2sin x + 1 = 0, subtracting 1 from both sides gives us 2sin x = -1. Dividing both sides by 2 gives us sin x = -1/2. From the unit circle, we can see that sin x = -1/2 at x = 210° or x = 330° (in degrees) or x = 7π/6 or x = 11π/6 (in radians).
By examining both equations, we find that the common solution is x = 330° or x = 11π/6. However, depending on the domain specified for x, there may be additional solutions based on the periodic nature of the sine and cosine functions.