Question

a+b+c+d=360 then cosa+cosb+cosc+cosd=

Answer 1

if a,b,c,d are angles in a cyclic quadrilateral,
then a+c = 180
b+d = 180
and that expression will = 0

so is it given that a,b,c,d are angles in a cyclic quadrilateral?

Answer 2

Bet that's an appropriate question, hartnn! But most of us on OpenStudy probably haven't heard the term "cyclic quadrilateral" before.

Answer 3

http://openstudy.com/study#/updates/4fc8abc6e4b022f1e13024db
I bet this is where hartnn got the idea from

Answer 4

going back to the original question
when a=b=c=d=90
cosa + cosb + cosc + cosd = 0

but when a=b=c=100 and d=60
cosa + cosb + cosc + cosd = -0.02...

So I was unable to define a general solution for cosa+cosb+cosc+cosd when a+b+c+d=360

Answer 5

Were I to experiment to determine whether there's a pattern here: I'd let each A, B, C and D equal 45 degrees and calculate cos A, cos B, cos C and cos D. Then I'd try other sets of angles whose sums are also 360 degrees. Not a fancy approach, but I'd expect it'd reveal some important facts/patterns here.