We are given the trigonometric equation sec(θ) = -3/2. We are to find all solutions of the equation in the interval [0, 2π).
The secant function, sec(θ), is the reciprocal of the cosine function, cos(θ). Therefore, we can rewrite our equation as 1/cos(θ) = -3/2.
To solve for cos(θ), we get the reciprocal of both sides resulting in cos(θ) = -2/3. Please note that the cosine function has a value between -1 and 1. Since it values on the unit circle, and -2/3 is within this range, it means there exists a solution to our equation.
Finding the angles that satisfy this equation involves solving for the values of θ that would yield a cosine value of -2/3 within the interval [0, 2π).
Solving for θ provides us with two solutions: 2.30052398302186, and 3.98266132415772.
Therefore, within the interval [0, 2π), the solutions for the equation secθ = -3/2 are θ = 2.30052398302186 and θ = 3.98266132415772.