Given that A,B,C be angles in an acute

Kiara Haas

Kiara Haas

Answered question

2022-03-24

Given that A,B,C be angles in an acute triangle.
If (5+4cosA)(54cosB)=9 and (1312cosB)(1312cosC)=25
find cos(A+C). I know A+B+C=180 and cosB=cos(A+C) and what next?

Answer & Explanation

undodaonePvopxl24

undodaonePvopxl24

Beginner2022-03-25Added 13 answers

The tangent half-angle substitution works really well here. Let a=tan(A2) , b=tan(B2) , c=tan(C2) then a,b,c>0 and
cosA=1a21+a2,cosB=1b21+b2,cosC=1c21+c2
Substituting these into the given equations and simplifying we (eventually) get
a=3b,b=15c
Using the triangle relationship we also have
tan(A2+B2)=tan(90C2)=cot(C2)
a+b1ab=1c
Thus
4b13b2=5b
b2=115
Back-substituting gives
cosB=cos(A+C)=78

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