Question 1 :
In a triangle ABC, prove that
(ii) a(cos B + cos C) = 2(b + c) sin2 A/2
Solution :
a(cos B + cos C) = 2(b + c) sin2 A
L.H.S
= a(cos B + cos C)
a/sin A = b/sin B = c/sin C = 2R
a = 2R sin A, b = 2R sin B and c = 2R sin C
= (2R sin A) [2cos (B+C)/2 cos (B-C)/2]
= (2R sin A) [2cos (180-A)/2 cos (B-C)/2]
= (2R sin A) [2cos (90-(A/2)) cos (B-C)/2]
= (2R sin A) [2 sin (A/2) cos (B-C)/2]
= 2R (2sin A/2 cos A/2) [2 sin (A/2) cos (B-C)/2]
= 2R sin2A/2 [4cos A/2 cos (B - C)/2]
= 2R sin2A/2 [4cos (180 - (B+C)/2) cos (B - C)/2]
= 2R sin2A/2 [4 sin (B+C)/2 cos (B - C)/2]
= 2R sin2A/2 [2 (2 sin (B+C)/2 cos (B - C)/2)]
= 2R sin2A/2 [2 (sin B + sin C)]
= 2 sin2A/2 [2R sin B + 2R sin C]
= 2 sin2A/2 [b + c] --> R.H.S
Hence proved.
(iii) (a2 − c2) / b2 = sin(A − C) / sin(A + C)
Solution :
L.H.S
= (a2 − c2) / b2
= [(2R sin A)2 - (2R sin C)2] / (2R sin B)2
= [4R2 sin2A - 4R2 sin2C] / (4R2 sin2B)
= [sin2A - sin2C] / sin2B
Formula for sin2A - sin2B = sin (A + B) sin (A - B)
= sin (A + C) sin (A - C) / sin2B
= sin (A + C) sin (A - C) / [sin(180 - (A + C)]2
= sin (A + C) sin (A - C) / [sin(A + C)]2
= sin (A - C)/sin (A + C)
R.H.S
Hence proved.
(iv) a sin(B − C)/(b2 − c2) = b sin(C − A)/c2 − a2 = c sin(A − B)/(a2 − b2)
Solution :
Part 1 :
= a sin(B − C)/(b2 − c2)
= (2R sin A) sin (B - C) / (2R sin B)2 - (2R sin C)2
= (2R sin A) sin (B - C) / (4R2 sin2 B - (4R2 sin2 C))
= a sin (B - C) /2R (sin2 B - sin2 C)
= a sin (B - C) / 2R sin (B+C) sin (B-C)
= a/2R sin (B + C)
= a/2R sin (180 - A)
= a/2R sin A
= a / a = 1 -----------(1)
Part 2 :
b sin(C − A)/c2 − a2
= (2R sin B) sin (C - A) / (2R sin C)2 - (2R sin A)2
= (2R sin B) sin (C - A) / (4R2 sin2 C - (4R2 sin2 A))
= b sin (C - A) /2R (sin2 C - sin2 A)
= b sin (C - A) / 2R sin (C+A) sin (C-A)
= b/2R sin (C + A)
= b/2R sin (180 - B)
= b/2R sin B
= b / b = 1 -----------(2)
Part 3 :
c sin(A − B)/a2 − b2
= (2R sin C) sin (A - B) / (2R sin A)2 - (2R sin B)2
= (2R sin C) sin (A - B) / (4R2 sin2 A - (4R2 sin2 B))
= c sin (A - B) /2R (sin2 A - sin2 B)
= c sin (A - B) / 2R sin (A+B) sin (A-B)
= c/2R sin (A + B)
= c/2R sin (180 - C)
= c/2R sin C
= c / c = 1 -----------(3)
(1) = (2) = (3)
Hence proved.
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