Balloon problem

[Zhowers]

So halfway through the problem I noticed that the problem could have been used as a law of cosines, and I feel like I might be wrong so far. I'm pretty stuck and would appreciate any help! Thank you.

 

[HallsofIvy]

Let the tension in the left rope be $\displaystyle T_1$ and the tension in the right rope be $\displaystyle T_2$. Draw a line straight down from the balloon. That divides the problem into two right triangles. On the left we have a hypotenuse of "length" $\displaystyle T_1$ and angle 60 degrees. The vertical component of force is $\displaystyle T_1 sin(60)$, downward, and horizontal component of force is $\displaystyle T_1 cos(60)$, to the left. On the right we have a hypotenuse of "length" $\displaystyle T_2$ and angle 25 degrees. The vertical component of force is $\displaystyle T_2 sin(25)$, downward, and horizontal component of force is $\displaystyle T_2 cos(25)$, to the right.

The total horizontal force must be 0: $\displaystyle T_1cos(60)= T_2cos(25)$. The total vertical force is also 0 but that exerted by the ropes must offset the upward lifting force, 570 pounds: $\displaystyle T_1sin(60)+ T_2sin(25)= 570$. Solve those two linear equations for $\displaystyle T_1$ and $\displaystyle T_2$.