Final answer:
To find the exact value of theta for the equation sec(θ) = -5/2 in Quadrant 3, we need to determine the reference angle of θ in Quadrant 1 and consider the signs of sin and cos. The exact value of θ for the given equation is θ = 210 degrees.
Step-by-step explanation:
To find the exact value of theta (θ) for the equation sec(θ) = -5/2 in Quadrant 3, we need to determine the reference angle of θ in Quadrant 1. The reference angle is the angle formed between the terminal side of θ and the x-axis. Since sec(θ) = -5/2, we know that cos(θ) = -2/5 because sec(θ) is the reciprocal of cos(θ). In Quadrant 1, cos(θ) is positive, so we take the positive value of cos(θ) which is 2/5.
Using the Pythagorean Identity, sin²(θ) + cos²(θ) = 1, we can find sin(θ) as follows:
sin²(θ) + (2/5)² = 1
sin²(θ) + 4/25 = 1
sin²(θ) = 21/25
sin(θ) = sqrt(21)/5
Since θ is in Quadrant 3, both sin(θ) and cos(θ) are negative. Therefore, we have sin(θ) = -sqrt(21)/5.
So, the exact value of θ for the equation sec(θ) = -5/2 with θ in Quadrant 3 is θ = 210 degrees (option c).