Right. I see how it would be (180-50) + t = 130 + t
but from here Idk how to use the cosine rule since both t and p will be unknown
TBH I'm not quite following KM's diagram, so let me try to explain how I would do this one by resolving into rectangular components.
First, I'd let the direction of i be parallel to the direction of the force of P newtons. So that force is P
i.
Second, I'd write the other two vectors in terms of their components. The 10 newtons force will be 10cos(50°)
i + 10sin(50°)
j, and the 12 newtons force will be 12cos(a)
i + 12sin(a)
j, where a is the angle measured anti-clockwise from the direction of the P newtons force. (There are other ways to resolve that 12 newtons force, by constructing some right-angled triangles which I think is what KM is suggesting, but this will do).
Then, since everything is in equilibrium, we have 10sin(50°) + 12sin(a) = 0 and P + 10cos(50°) + 12cos(a) = 0. Solve simultaneously for a and P. Note that this approach will give two solutions for a (since you'll get sin(a) = something negative), but only one of these corresponds to a positive solution for P.
Using this approach, the angle between the 12 newtons and 10 newtons forces will be a - 50.
If you use the cosine rule, then draw a triangle where the side lengths are 12, 10, P, and the angle between the sides of lengths 10 and P is (180 – 50)°, now use cosine rule to find P.