Question

suppose a triangle has two sides of lengths a and b. if the angle between these sides varies, what is the maximum possible area that the triangle can attain? what can you say about the minimum possible area?

Answer 1

A = (1/2)(ab)sinØ , where Ø is the angle between the two fixed sides a and ab
dA/dØ = (1/2)(ab)cosØ
= 0 for a max of A
then
cosØ = 0
Ø = 90°

the maximum area = (1/2)(ab)sin90° = (1/2)ab
(which of course is the popular formula for the area of a right-angled triangle. )

Common sense will tell us that the minimum area will be obtained when the angle is 0, that is, there is no triangle.

Answer 2

i don't understand where did the d come from and how did you turn it into cos?