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?

1 Answer
Dec 16, 2017

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#

enter image source here

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#