EXAMPLE PROBLEMS FOR LAW OF SINES AND COSINES

About "Example Problems for Law of Sines and Cosines"

Example Problems for Law of Sines and Cosines :

Here we are going to see some example problems on law of sines and cosines.

Example Problems for 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 - 2sin2- 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

2b2/4R=  (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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