For triangle ABC, find maximum value of 8 + sin ⁡ 2 A + sin...

Patatiniuh

Patatiniuh

Answered

2022-07-05

For triangle ABC, find maximum value of 8 + sin 2 A + sin 2 B + sin 2 C sin A + sin B + sin C
Considering only the right hand part of the expression:
sin 2 A + sin 2 B + sin 2 C sin A + sin B + sin C
= 4 sin A sin B sin C sin A + sin B + sin C

Answer & Explanation

Kiana Cantu

Kiana Cantu

Expert

2022-07-06Added 22 answers

Note that the area of the triangle A r e a = a b c 4 R = 1 2 r ( a + b + c ). Then, per the sine rule
4 sin A sin B sin C sin A + sin B + sin C = 4 a b c 8 R 3 a + b + c 2 R = 2 r R 1
where, given the circumradius R, the inradius is the largest for equilateral triangle.

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