Question 339074
cosecant = hypotenuse/opposite

180 < theta < 270 tells me that we are in quadrant 3.  

We form a right triangle in quadrant 3.  The hypotenuse = -5 and the leg opposite the hypotenuse = 4.  We are looking for the adjacent side of the right triangle.

Let A = adjacent side

A^2 + 4^2 = (-5)^2

A^2 + 16 = 25

A^2 = 25 - 16

A^2 = 9

sqrt[A^2] = sqrt[9]

A = 3

tangent = opposite side/adjacent

In quadrant 3, tangent is positive.

Answer: tan(theta) = 4/3...CHOICE (C).