Show that: sinA + sinB + sinc > cosA + cosB + cosc,?

Question

if ABC is an acute angled triangle

Answer 1

simple..
do this
as A,B,C is an acute triangle this implies all angles are less than 90.

Now well start...
fr a triangle u hv a property tht
1>cosA + cosB + cosc>1.5
and
1.5>sinA + sinB + sinC>2
Hence ur result follows..


the proof of these
here

cosA + cosB + cosC
=2cos((A+B)/2).cos((A-B)/2) + 1 - 2sin^2 (C/2)
=1 + 2sin(c/2)cos(A-B /2) - 2sin^2 (C/2)
=1 + 2sin(c/2){cos(A-B /2) - sin (C/2)}
=1 + 2sin(c/2){cos(A-B /2) - cos (A+B/2)}
=1 + 4sinC/2 sinB/2 sinA/2
as
1>sinC/2 sinB/2 sinA/2>0
this implies
1>cosA + cosB + cosC>1.5

u can proof the other similarly!

Answer 2

yes this possible SIN(A+B)= SINACOSB+COSASINB SAME way u put the equation of SIN(A+B+C) and compare you will get the result still if you have any maths doubts then contact me on my mobile no 09824776781 but you have to pay me 100 rs for each question

Answer 3

sinA+sinB+sinC solve ho gaya na

Answer 4

For ? ABC, A + B + C = ? => A/2 + B/2 + C/2 = ?/2 => A/2 + B/2 = ?/2 - C/2 => tan(A/2 + B/2) = tan(?/2 - C/2) => [tan(A/2) + tan(B/2)] / [one million - tan(A/2) tan(B/2)] = cot(C/2) = one million/tan(C/2) => tan(C/2) tan(A/2) + tan(B/2) tan(C/2) = one million - tan(A/2) tan(B/2) => tan(A/2) tan(B/2) + tan(B/2) tan(C/2) + tan(C/2) tan(A/2) = one million ... ( one million ) additionally, A/2 + B/2 + C/2 = ?/2 => sin(A/2 + B/2 + C/2) = sin(?/2) => sin(A/2) cos(B/2) cos(C/2) + sin(B/2) cos(C/2) cos(A/2) + sin(C/2) cos(A/2) cos(B/2) - sin(A/2) sin(B/2) sin(C/2) = one million ... ( 2 ) From ( one million ) and ( 2 ), sin(A/2) cos(B/2) cos(C/2) + sin(B/2) cos(C/2) cos(A/2) + sin(C/2) cos(A/2) cos(B/2) - sin(A/2) sin(B/2) sin(C/2) = tan(A/2) tan(B/2) + tan(B/2) tan(C/2) + tan(C/2) tan(A/2) => sin(A/2) cos(B/2) cos(C/2) + sin(B/2) cos(C/2) cos(A/2) + sin(C/2) cos(A/2) cos(B/2) = sin(A/2) sin(B/2) sin(C/2) + tan(A/2) tan(B/2) + tan(B/2) tan(C/2) + tan(C/2) tan(A/2).

Answer 5

hellllloooooo.........u can use the trigo rules to solve tht......

Answer 6

can u please provide more details