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 \(\displaystyle T_2\).
