The sign is held in equilibrium. Using Newton's second law, we set up the equations of the net forces acting on the sign in the horizontal and vertical directions:
∑ F (horizontal) = T₁ cos(29.5°) - T₂ cos(44.5°) = 0
(right is positive, left is negative)
∑ F (vertical) = T₁ sin(29.5°) + T₂ sin(44.5°) - 215 N = 0
(up is positive, down is negative)
Solve the system of equations. I use elimination here:
• Multiply the first equation by sin(29.5°) and the second by cos(29.5°):
sin(29.5°) (T₁ cos(29.5°) - T₂ cos(44.5°)) = 0
cos(29.5°) (T₁ sin(29.5°) + T₂ sin(44.5°) - 215 N) = 0
T₁ cos(29.5°) sin(29.5°) - T₂ cos(44.5°) sin(29.5°) = 0
T₁ cos(29.5°) sin(29.5°) + T₂ cos(29.5°) sin(44.5°) = (215 N) cos(29.5°)
• Subtract the first equation from the second to eliminate T₁ :
T₂ cos(29.5°) sin(44.5°) - (- T₂ cos(44.5°) sin(29.5°)) = (215 N) cos(29.5°)
• Solve for T₂ :
T₂ (cos(29.5°) sin(44.5°) + cos(44.5°) sin(29.5°)) = (215 N) cos(29.5°)
T₂ sin(74.0°) = (215 N) cos(29.5°)
… … … (using the fact that sin(x + y) = sin(x) cos(y) + cos(y) sin(x))
T₂ = (215 N) cos(29.5°) / sin(74.0°)
T₂ ≈ 195 N
• Solve for T₁ :
T₁ cos(29.5°) - T₂ cos(44.5°) = 0
T₁ cos(29.5°) = T₂ cos(44.5°)
T₁ = T₂ cos(44.5°) / cos(29.5°)
T₁ ≈ 160. N
