Final answer:
After rewriting the equation 2 cos θ + 3 = - sec θ in terms of cosine and factoring, we find that θ = 120° and θ = 240° are the solutions over the interval [0°, 360°), which are not listed in the provided options.
Step-by-step explanation:
To solve the equation for solutions over the interval [0°, 360°), given by 2 cos θ + 3 = - sec θ, where θ is the angle in degrees, we need to work within the trigonometric identities. The secant function is the reciprocal of the cosine function, so we begin by rewriting the equation with this relationship in mind:
2 cos θ + 3 = -1/cos θ
To find common denominators and solve the equation, we multiply both sides by cos θ:
2 cos^2 θ + 3 cos θ = -1
Next, we set this equation to zero to solve for cos θ:
2 cos^2 θ + 3 cos θ + 1 = 0
However, this is a quadratic equation in terms of cos θ, which we can factor:
2 cos θ + 1)(cos θ + 1) = 0
Setting each factor equal to zero gives us:
2 cos θ + 1 = 0
cos θ + 1 = 0
These equations lead to:
cos θ = -1/2
cos θ = -1
Considering the interval [0°, 360°), the angles that satisfy these conditions are:
θ = 120° and θ = 240° for cos θ = -1/2
θ = 180° for cos θ = -1
None of the provided answers A) θ = 0°, B) θ = 45°, C) θ = 90°, or D) θ = 180° are correct. The angles that are solutions to the original equation are 120° and 240°. Therefore, the correct answer is not listed among the options given.