Verified answer

I mistakenly solved for the angle between the 8N and 10N forces, but at the bottom I have the solution for the angle between the 8N and 9N forces.

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"Lami's Theorem" is a corollary of the Law of Sines.  Anyway it says that

the ratio of each force to the sine of the angle between the other two forces is the same ratio.  So you have

10/sin(C) = 9/sin(B) = 8/sin(A), where A is the angle opposite the 8N force, B is the angle opposite the 9N force, and C opposite the 10N force.

Now suppose you had a triangle whose sides were 8, 9, and 10 units long.  You would have exactly the same set of equations as are shown above, and you could solve the triangle with the Law of Cosines:

9^2 = 10^2 + 8^2 - 2*8*10*cos(B) =>

cos(B) = (100 + 64 - 81)/160 = 0.51875 => B = 58.75 degrees.

However, in the original problem, the angle B will be BETWEEN 90 and 180, so it's not 58.75 degrees, it's 121.25 degrees.

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To check that this works, I note that in the triangle version, I have

cos(A) = (100 + 81 - 64)/180 = 0.65 => A = 49.46 degrees, and in the concurrent-forces problem I expect A = 130.54 degrees.  If so, then angle C should be 360 - 121.25 - 130.54 = 108.21 degrees.

Let's check 10/sin(108.21) = 9/sin(121.25) = 8/sin(130.54). I get

10.527 = 10.527 = 10.527.  Because I rounded off the angle measures, I get slight differences in these ratios at the 6th significant digit.  No problem.

Final answer: the angle you actually asked for in your question is the 108.21 degrees.

Lami's theorem

Vectors keeping an object in static equilibrium, with the angles directly opposite to the corresponding vectors.

A/sinα = B/sinβ = C/sinγ

α + β + γ = 360º

where A, B and C are the magnitudes of the three coplanar, concurrent and non-collinear vectors, VA, VB, VC, which keep the object in static equilibrium, and α, β and γ are the angles directly opposite to the vectors.

A = 8, B = 9, C = 10 N

α is angle between B and C

β is angle between A and C

γ is angle between B and A  (desired angle)

α + β + γ = 360º

10/sinγ = 8/sinα = 9/sinβ

three eq in 3 unknowns, in theory solvable.