**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 a^{2}, b^{2}, c^{2} 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(1-2sin^{2}B) = 1 - 2sin^{2}A + 1 - 2sin^{2}C

2 - 4sin^{2}B = 2 - 2sin^{2}A - 2sin^{2}C

4sin^{2}B = 2sin^{2}A + 2sin^{2}C

2sin^{2}B = sin^{2}A + sin^{2}C

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

2sin^{2}B = sin^{2}A + sin^{2}C

2(b/2R)^{2 = }(a/2R)^{2 }+ (c/2R)^{2}

2b^{2}/4R^{2 }= (a^{2} + c^{2})/4R^{2}

2b^{2 }= (a^{2} + c^{2})

Hence a^{2}, b^{2 }and c^{2} 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

After having gone through the stuff given above, we hope that the students would have understood, "Example Problems for Law of Sines and Cosines"

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