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ABC-with-AB-5-BC-7-CA-8-Find-the-value-of-sin-A-sin-B-sin-C-cot-A-2-cot-B-2-cot-C-2-




Question Number 10363 by Joel575 last updated on 05/Feb/17
ΔABC with AB = 5, BC = 7, CA = 8  Find the value of  (sin A + sin B + sin C) . (cot (A/2) + cot (B/2) + cot (C/2))
ΔABCwithAB=5,BC=7,CA=8Findthevalueof(sinA+sinB+sinC).(cotA2+cotB2+cotC2)
Answered by mrW1 last updated on 06/Feb/17
a=BC=7  b=AC=8  c=AB=5  s=((a+b+c)/2)=((7+8+5)/2)=10  t=(√(s(s−a)(s−b)(s−c)))=(√(10×3×2×5))=10(√3)  sin A=((2t)/(bc))=((2×10(√3))/(8×5))=((√3)/2)  cos A=(√(1−(3/4)))=(1/2)  sin B=((2t)/(ac))=((2×10(√3))/(7×5))=((4(√3))/7)  cos B=(√(1−((48)/(49))))=(1/7)  sin C=((2t)/(ba))=((2×10(√3))/(8×7))=((5(√3))/(14))  cos C=(√(1−((75)/(196))))=((11)/(14))  cot (A/2)=((1+cos A)/(sin A))=((1+(1/2))/((√3)/2))=(3/( (√3)))  cot (B/2)=((1+cos B)/(sin B))=((1+(1/7))/((4(√3))/7))=(2/( (√3)))  cot (C/2)=((1+cos C)/(sin C))=((1+((11)/(14)))/((5(√3))/(14)))=(5/( (√3)))  (sin A+sin B+sin C)(cot (A/2)+cot (B/2)+cot (C/2))  =(((√3)/2)+((4(√3))/7)+((5(√3))/(14)))((3/( (√3)))+(2/( (√3)))+(5/( (√3))))  =((20(√3))/(14))×((10)/( (√3)))=((100)/7)
a=BC=7b=AC=8c=AB=5s=a+b+c2=7+8+52=10t=s(sa)(sb)(sc)=10×3×2×5=103sinA=2tbc=2×1038×5=32cosA=134=12sinB=2tac=2×1037×5=437cosB=14849=17sinC=2tba=2×1038×7=5314cosC=175196=1114cotA2=1+cosAsinA=1+1232=33cotB2=1+cosBsinB=1+17437=23cotC2=1+cosCsinC=1+11145314=53(sinA+sinB+sinC)(cotA2+cotB2+cotC2)=(32+437+5314)(33+23+53)=20314×103=1007
Commented by Joel575 last updated on 06/Feb/17
thank you sir
thankyousir

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