Question

in triangle abc , abc sin a/2 ,sin b/2,sin c/2=​

Answer 1

Answer:

sin a/2, sin b/2 , sin c/2 ≤ 1/8

Step-by-step explanation:

We know arithmetic mean is always greater than or equal to geometric mean.

therefore. (sin^2(A/2) sin^2(B/2) sin^2(C/2))⅓ ≤ 1/3 (sin^2(A/2) + sin^2(B/2) + sin^2(C/2))

Also we know,

(sin^2(A/2) + sin^2(B/2) + sin^2(C/2)) ≥ 3/4

therefore

(sin^2(A/2) sin^2(B/2) sin^2(C/2))⅓ ≤ 1/3 (3/4)

=> (sin^2(A/2) sin^2(B/2) sin^2(C/2))⅓ ≤ 1/4

=> (sin^2(A/2) sin^2(B/2) sin^2(C/2)) ≤ (1/4)³

=> (sin(A/2) sin(B/2) sin(C/2)) ≤ (1/8)