MHB Applying Trigonometric Functions

[soccerstar814]

Okay, so I know how to apply basic trigonometric functions to solve right triangles. However, I am not quite sure how to manipulate trigonometric functions to solve for more complex questions involving non-right triangles, like the question I have attached. I have to use the basic trigonometric functions only to solve this question (no sine law or cosine law, etc.) My objective is to find the length of line segment TU. Any help would be greatly appreciated! Thanks.

 

[MarkFL]

I would let $x=\overline{UV}$ and $y=\overline{TU}$ then state:

$$\tan(53.8^{\circ})=\frac{y}{x}$$

$$\tan(38.8^{\circ})=\frac{y}{x+12.5}$$

Now you have two equations in two unknowns...