The minimum of cos (a) + cos (b) + cos (c)

[Guest]

Given a triangle with angles a,b,c.
The fact that [tex]cos (a) + cos (b) + cos (c) <= 3/2[/tex], can be found at http://mathforum.org/library/drmath/view/54120.html
What is known about the minimum of [tex]cos (a) + cos (b) + cos (c)[/tex] ?

[Guest]

You can use the identity
cos a + cos b + cos c = 1 + 4 sin(a/2) sin(b/2) sin(c/2)
when a+b+c = 180 degrees.
(For a proof see https://sg.answers.yahoo.com/question/index?qid=20091218205518AAtsWzf or just google it.)

Because a,b,c are angles of a triangle sin(a/2) sin(b/2) sin(c/2) is non-negative. So the minimum value of cos a + cos b + cos c is 1, which occurs when one of the angles is zero. If you don't like the idea of triangles having angles which are 0, then you would say cos a + cos b + cos c > 1 and we can get arbitrarily close to 1 by making one of the angles sufficiently close to 0.

Hope this helped,

R. Baber.