MHB Solving for AB & AC: Find the Solution

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The discussion revolves around solving for the lengths AB and AC in a physics problem involving forces and equilibrium. Two key equations are established based on the sine and cosine components of the forces, leading to a system of equations. The user expresses uncertainty about missing information, particularly the value of BC, which is crucial for solving the equations. The conversation highlights the need to resolve forces into their x and y components to apply equilibrium conditions effectively. Ultimately, the user seeks assistance in finding the values of AB and AC based on the established equations.
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Find AB and AC in the figure given.

I came up with 2 equations which have 3 unknowns,

$AB\sin(55)+AC\sin(30)=BC$
$AB\sin(35)=AC\sin(60)$

I'm not sure if I'm just missing some given in the problem. Because if BC is given I can solve the equation above. Please help. thanks!
 

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First, let's resolve the 3 forces into their $x$ and $y$ components (where point $D$ is directly below $A$, the mass):

[table="width: 400, class: grid, align: left"]
[tr]
[td]Force[/td]
[td]$x$-component[/td]
[td]$y$-component[/td]
[/tr]
[tr]
[td]$\vec{AB}$[/td]
[td]$-\vec{AB}\sin\left(55^{\circ}\right)$[/td]
[td]$\vec{AB}\cos\left(55^{\circ}\right)$[/td]
[/tr]
[tr]
[td]$\vec{AC}$[/td]
[td]$\vec{AC}\sin\left(30^{\circ}\right)$[/td]
[td]$\vec{AC}\cos\left(30^{\circ}\right)$[/td]
[/tr]
[tr]
[td]$\vec{AD}$[/td]
[td]$0$[/td]
[td]$-360\text{ N}$[/td]
[/tr]
[/table]

Now, the first condition for equilibrium gives us the equations:

$$\sum F_x=\vec{AC}\sin\left(30^{\circ}\right)-\vec{AB}\sin\left(55^{\circ}\right)=0$$

$$\sum F_y=\vec{AB}\cos\left(55^{\circ}\right)+\vec{AC}\cos\left(30^{\circ}\right)-360\text{ N}=0$$

Now, you have two equations in two unknowns. :D
 
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