a+b+c=180° prove that cosa + cosb + cosc

Tahmid Knox 2021-01-30 Answered

a+b+c=180 prove that cosa+cosb+cosc

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Expert Answer

Gennenzip
Answered 2021-01-31 Author has 96 answers

If a=b=c=60º,cosa+cosb+cosc=312=1.5.
cosc=cos(180(a+b))=cos(a+b).
Let

a=60aandb

=60+afor0a<60º,cos(0)=1,cos(60)=0.5.
Trig identity:

cos(AB)+cos(A+B)

=cosAcos+sinAsinB+cosAcosBsinAsinB=2cosAcosB.
We have:

cos(60a)+cos(60+a)cos(120)

=2cos(60)cos(a)+0.5=cos(a)+0.5.1.5.
Also, as a+b approaches 0, the expression on the left <1+11<1. 1 is less than 1.5.
As a+b approaches 180, the expression becomes cosa+cosb+1.
Let a=90a and b=90+a,

0cos(90a)+cos(90+a)+1=2cos90cos(a)+1=1

less than 1.5.

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