Example Problems for Law of Sines and Cosines :
Here we are going to see some example problems on law of sines and cosines.
Question 1 :
In a triangle ABC, if sin A/sin C = sin(A − B)/sin(B − C), prove that a2, b2, c2 are in Arithmetic Progression.
Solution :
sin A/sin C = sin(A − B)/sin(B − C)
sin A sin (B - C) = sin C sin (A - B)
Since A, B and C are angles of triangle
A + B + C = 180
A = 180 - (B + C)
C = 180 - (A + B)
sin A sin (B - C) = sin C sin (A - B)
sin (180 - (B+C)) sin (B-C) = sin (180 - (A+B)) sin (A-B)
sin (B+C) sin (B-C) = sin (A+B) sin (A-B)
(1/2)[cos (B+C-B+C) - cos (B+C+B-C)] = (1/2)[cos (A+B-A+B) - cos (A+B+A-B)]
(1/2)[cos 2C - cos 2B] = (1/2)[cos 2B - cos 2A]
cos 2C - cos 2B = cos 2B - cos 2A
2 cos 2B = cos 2A + cos 2C
2(1-2sin2B) = 1 - 2sin2A + 1 - 2sin2C
2 - 4sin2B = 2 - 2sin2A - 2sin2C
4sin2B = 2sin2A + 2sin2C
2sin2B = sin2A + sin2C
Using the law of sin, we get
a/sin A = b/sin B = c/sin C = 2R
sin A = a/2R, sin B = b/2R, sin C = c/2R
2sin2B = sin2A + sin2C
2(b/2R)2 = (a/2R)2 + (c/2R)2
2b2/4R2 = (a2 + c2)/4R2
2b2 = (a2 + c2)
Hence a2, b2 and c2 are in A.P
Question 2 :
The angles of a triangle ABC, are in Arithmetic Progression and if b : c = √3 : √2, find ∠A.
Solution :
Let A, B and C be the angles of a triangle.
Since the angles are in arithmetic progression, the following relationship holds.
B - A = C - B
2B = A + C
A + B + C = 180
2B + B = 180
3B = 180, B = 60
Laws of sine :
a/sin A = b/sin B = c/ sin C = 2R
a/sin A = √3/sin 60 = √2/sin C
a/sin A = √3/(√3/2) = √2/sin C
2 = √2/sin C
sin C = 1/√2
C = 45
A + 60 + 45 = 180
A + 105 = 180
A = 75
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