Question

Is there any formula to find the third side of a triangle with any angle between the sides,Pythagoras theorm works only for right angled triangles.If yes what is it?

Answer 1

See below.

Explanation:

Think of two non null vectors `vec u, vec v` representing two triangle sides making an angle `phi` between them. the third side can be represented as `vec w = vec u - vec v` with length

`norm(vec w)^2 = norm(vec v - vec u)^2` or

`norm(vec w)^2 = norm(vec v)^2+norm(vec u)^2-2 << vec u, vec v >>`

but `norm vecw =c, norm vec u=a` and `norm vec v = b` and finally

`c^2=a^2+b^2-2a b cos phi`

NOTE

`vec u = (u_1,u_2)`
`norm(vec u) = sqrt(u_1^2+u_2^2)`
`<< vec u, vec v >> = u_1v_1+u_2v_2 = norm(vec u)norm(vec v) cos phi`