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))

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)

Commented byJoel575 last updated on 06/Feb/17

thank you sir