a diagram of a bike frame. The frame can be approximated as two triangles that have the seat tube as a common side.

Question

Suppose the frame has a top tube length of 52.3 centimeters, a seat tube length of 48.0 centimeters, and a seat angle of 70.0°. Find the length of the down tube and the angle between the seat tube and the down tube. 

Answer 1

a=48

b=52.3

γ=70°

According to your words; you're asking for the diagonal which is side c.

β is at the bottom.

To get c; a^2+b^2-2ab(cosγ)=c^2

48^2+52.3^2-2*48*52.3*.342=c^2

2304+2735- 5021* .342=c^2

5039.29- 1717.21 =c^2

3322.08 ^.5=57.637=c

c/sinγ=b/sinβ

57.637/sin70=b/sin52.3

=61.336 =b/.791

b=48.531