Final answer:
To solve the trigonometric equation 2 cos² x + 5 cos x + 2 = 0 in the interval [0,2π), substitute cos x as y, factor the quadratic equation, and solve for x.
Step-by-step explanation:
To solve the trigonometric equation 2 cos² x + 5 cos x + 2 = 0 in the interval [0,2π), we can use a substitution to solve it as a quadratic equation. Let's substitute cos x as y:
2y² + 5y + 2 = 0
Now we can factor the quadratic equation:
(2y + 1)(y + 2) = 0
Solving for y:
2y + 1 = 0 or y + 2 = 0
From here, we can solve for x:
2 cos x + 1 = 0 or cos x + 2 = 0
Solving for x:
cos x = -1/2 or cos x = -2
Since cos x cannot be -2, we focus on solving cos x = -1/2:
x = π/3 or x = 5π/3