Question

What values for θ (0<=θ<=2π) satisfy the equation? 2sin θ cos θ + √3 cos θ=0

Answer 1

[AI-generated answer — Bot GPT 3.5]

To solve the equation 2sin(θ)cos(θ) + √3cos(θ) = 0, we can first factor out a common cos(θ) term:

cos(θ)(2sin(θ) + √3) = 0

Now, we have two possibilities:

1. cos(θ) = 0
This occurs when θ = π/2 and 3π/2.

2. 2sin(θ) + √3 = 0
Subtract √3 from both sides:
2sin(θ) = -√3
Divide by 2:
sin(θ) = -√3/2

This occurs in the third and fourth quadrants, so we have two additional solutions:
θ = 5π/3 and θ = 7π/3

Therefore, the values for θ in the interval [0, 2π] that satisfy the equation are:
θ = π/2, 3π/2, 5π/3, and 7π/3.

Answer 2

wrong, since 7π/3 > 2π

so you want
θ = π/2, 3π/2, 4π/3, and 5π/3.

Answer 3

[AI-generated answer — Bot GPT 3.5]

Apologies for the oversight. You are correct. The correct values for θ that satisfy the equation 2sin(θ)cos(θ) + √3cos(θ) = 0 in the interval [0, 2π] are:

θ = π/2, 3π/2, 4π/3, and 5π/3. Thank you for pointing that out.