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Find the exact value of θ for the equation sec(θ) = -5/2, with θ in Quadrant 3.

a) θ = 120 degrees
b) θ = 150 degrees
c) θ = 210 degrees
d) θ = 240 degrees

User Brisi
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1 Answer

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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).

User Bofredo
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