Final answer:
The question presents a physics problem involving tensions and angles in which equations are manipulated to solve for an unknown variable. By substituting expressions and applying trigonometric identities, the problem is simplified to a solvable state.
Step-by-step explanation:
The initial question refers to an equation that is typical of a problem involving physics, specifically dynamics and forces. The given equation T₁ sin (30°)+T₂ sin (45°)= w indicates a system with two tensions at different angles balancing a weight. By substituting the value of T₂ in terms of T₁, T₂ = (1.225)T₁, you simplify the equation to solve for the single unknown, T₁. The same principle applies when solving for an unknown in a momentum equation involving cosine and sine components of velocity, m₁v₁ = m₂v'₂ cos θ₂ and m₁v₁ sin θ₁ + m₂v'₂ sin θ₂. Taking the ratio of these components and applying trigonometric identities, such as tan θ = sin θ / cos θ, can resolve an equation with a single unknown quantity.
In the context of physics, it is common to encounter problems requiring the manipulation of equations and the application of trigonometric identities to isolate and solve for unknown variables. These problems often involve understanding the relationship between forces, angles, and resultant motions. Notably, when the angles involved are not equal, the tensions will differ due to the directional components of the forces acting at different angles.