Evaluating \frac{\sin A + \sin B + \sin C}{\cos A

iocasq4

iocasq4

Answered question

2022-01-29

Evaluating sinA+sinB+sinCcosA+cosB+cosC for a triangle with sides 2, 3, 4

Answer & Explanation

Emilie Booker

Emilie Booker

Beginner2022-01-30Added 14 answers

There are well-known identities for ABC with the angles A,B,C, sides a,b,c, semiperimeter ρ=12a+b+c area S, radius r of inscribed and radius R of circumscribed circles,
sinA+sinB+sinC=ρR,
cosA+cosB+cosC=r+RR
so
x=sinA+sinB+sinCcosA+cosB+cosC=ρr+R
we also know that
R=abc4S,
r=Sρ,
S=144(ab)2(a2+b2c2)2
thus we can find that for a=2,b=3,c=4
ρ=92
S=3154
R=81515
r=156
x=ρr+R=31571.6598500

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