Final answer:
Using trigonometric identities and the given tan(θ), we can calculate sin(θ) and cos(θ), thus finding sec(θ) as 1/cos(θ) to determine the values of 'X' and 'Y' for the ratio sec(θ) = X/Y.
Step-by-step explanation:
To determine the values of 'X' and 'Y' that satisfy the ratio sec(θ) = X/Y, given that tan(θ) = 4√10/18, we can use trigonometric identities. We know that:
- sec(θ) = 1/cos(θ)
- tan(θ) = sin(θ)/cos(θ)
Since tan(θ) is given, we can express sin(θ) and cos(θ) in terms of this value. We have:
tan(θ) = sin(θ)/cos(θ) = 4√10/18
We can write sin(θ) = 4√10 * cos(θ) / 18, and since sin2(θ) + cos2(θ) = 1, we can find cos(θ) and sin(θ).
Once cos(θ) is known, sec(θ) can be easily calculated as 1/cos(θ), thus giving us the values for X and Y as the numerator and denominator of the fraction representing sec(θ).