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

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

Apart from the stuff given in "Example Problems for Law of Sines and Cosines", if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. SAT Math Videos

    May 22, 24 06:32 AM

    sattriangle1.png
    SAT Math Videos (Part 1 - No Calculator)

    Read More

  2. Simplifying Algebraic Expressions with Fractional Coefficients

    May 17, 24 08:12 AM

    Simplifying Algebraic Expressions with Fractional Coefficients

    Read More

  3. The Mean Value Theorem Worksheet

    May 14, 24 08:53 AM

    tutoring.png
    The Mean Value Theorem Worksheet

    Read More