Final answer:
The value of 3secθ+5tanθ is -5/3.
Step-by-step explanation:
To find the value of 3secθ+5tanθ when tanθ = -4/3 and (3π/2 < θ < 2π), we need to first find the value of secθ and tanθ. Since tanθ = -4/3, we can determine that the opposite side is -4 and the adjacent side is 3.
Using the Pythagorean theorem, we can find the hypotenuse as √((-4)^2 + 3^2) = 5.
Therefore, secθ = 1/cosθ = hypotenuse/adjacent = 5/3. Now, substituting the values of secθ and tanθ into the expression 3secθ+5tanθ, we get 3(5/3) + 5(-4/3) = 5 - 20/3 = -5/3.