The forces are balanced in the vertical direction.

Therefore:

$\overline{){\mathbf{\Sigma}}{\mathbf{F}}{\mathbf{=}}{\mathbf{0}}}$

${\mathit{T}}_{\mathbf{1}}\mathit{s}\mathit{i}\mathit{n}{\mathit{\theta}}_{\mathbf{1}}\mathbf{+}{\mathit{T}}_{\mathbf{2}}\mathit{s}\mathit{i}\mathit{n}{\mathit{\theta}}_{\mathbf{2}}\mathbf{=}\mathit{m}\mathit{g}$

The forces are also balanced in the horizontal direction.

F_{net} = 0

${\mathit{T}}_{\mathbf{1}}\mathit{c}\mathit{o}\mathit{s}{\mathit{\theta}}_{\mathbf{1}}\mathbf{=}{\mathit{T}}_{\mathbf{2}}\mathit{s}\mathit{i}\mathit{n}{\mathit{\theta}}_{\mathbf{2}}$

${\mathit{T}}_{\mathbf{2}}\mathbf{=}{\mathit{T}}_{\mathbf{1}}\mathbf{\left(}\mathit{c}\mathit{o}\mathit{s}{\mathit{\theta}}_{\mathbf{1}}\mathbf{\right)}\mathbf{/}\mathbf{\left(}\mathit{c}\mathit{o}\mathit{s}{\mathit{\theta}}_{\mathbf{2}}\mathbf{\right)}$

Hanging Chandelier (Figure 1 A chandelier with mass m is attached to the ceiling of a large concert hall by two cables. Because the ceiling is covered with intricate architectural decorations (not indicated in the figure, which uses a humbler depiction), the workers who hung the chandelier couldn't attach the cables to the ceiling directly above the chandelier Instead, they attached the cables to the ceiling near the walls. Cable 1 has tension T_{i} and makes an angle θ_{1} with the ceiling. Cable 2 has tension T_{2} and makes an angle θ_{2} with the ceiling.

Find an expression for *T*_{1}, the tension in cable 1, that does not depend on *T*_{2}.

Express your answer in terms of some or all of the variables *m*, *θ*_{1}, and *θ*_{2}, as well as the magnitude of the acceleration due to gravity *g*. You must use parentheses around *θ*_{1} and *θ*_{2}, when they are used as arguments to any trigonometric functions in your answer.

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