TRIGONOMETRY: THE RULE OF COSINES IN TRIANGLES

Law of Cosines

The law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. Using trigonometry, we can now obtain values of distances and angles which cannot be measured otherwise. The law of cosines finds application while computing the third side of a triangle given two sides and their enclosed angle, and for computing the angles of a triangle if all three sides are known to us.

 

A triangle has 6 elements (3 sides + 3 angles). Let us understand the law of the cosines formula and its derivation to study the inter-relationship of these elements using the cosine function.

 

What is Law of Cosines?

The law of cosine helps in establishing a relationship between the lengths of sides of a triangle and the cosine of its angles. The cosine law in trigonometry generalizes the Pythagoras theorem, which applies to a right triangle.

 

Law of Cosines: Definition

Statement: The law of cosine states that the square of any one side of a triangle is equal to the difference between the sum of squares of the other two sides and double the product of other sides and cosine angle included between them.

Let a, b, and c be the lengths of the three sides of a triangle and A, B, and C be the three angles of the triangle. Then, the law of cosine states that: a2 = b2 + c2 − 2bc·cosA. As stated above, the law of cosines in trigonometry generalizes the Pythagorean theorem. If you plug 90º for the angle in one of the rules, what will happen? Since cos 90º = 0, we are left with the Pythagoras theorem.

The law of cosine is also known as the cosine rule. This law is useful to find the missing information in any triangle. For example, if you know the lengths of two sides of a triangle and an angle included between them, this rule helps to find the third side of the triangle. Let us check out different cosine law formulas and the method to find these missing parameters in the following sections.

 

Law of Cosines Formula

The law of cosines formula can be used to find the missing side of a triangle when its two sides and the included angle is given i.e., it is used in the case of a SAS triangle. We know that if A, B, and C are the vertices of a triangle, then their opposite sides are represented by the small letters a, b, and c respectively. The law of cosines formula is used to:

 

1. find a when b, c, and A are given (or)

2. find b when a, c, and B are given (or)

3. find c when a, b, and C are given (or)

4. find any angle of the triangle when a, b, and c are given.

There are three laws of cosines and we choose one of them to solve our problems depending on the available data.

a² = b² + c² - 2bc·cos A

b² = c² + a² - 2ac·cos B

c² = a² + b² - 2ab·cos C

 

For more details, pay attention to the following examples.

 

Question 1

It is known that triangle ABC has sides AB = 6 cm, AC = 4 cm and angle BAC = 60o. Find the length of side BC.

Answer :

Triangle ABC can be described as follows. 

To determine the length of BC use the following formula.

BC² = AB² + AC² - 2AB · AC · cos A

BC² = 6² + 4² - 2 · 6 · 4 · cos 60^o

       = 36 + 16 - 48 · (1/2)

       = 52 - 24

       = 28

BC =   =  

So, the length of side BC is  cm.

 

Question 2

It is known that triangle ABC has side lengths AB = 6 cm, AC = 5 and BC = 4 cm. Determine the angle BCA.

Answer :

Triangle ABC can be described as follows.

To determine the angle BCA we use the following formula.

AB² = AC² + BC² - 2AC · BC · cos BCA

6² = 5² + 4² - 2 · 5 · 4 · cos BCA

36 = 25 + 16 - 40 · cos BCA

36 = 41 - 40 · cos BCA

   40 · cos BCA = 41 - 36

   40 · cos BCA = 5

          cos BCA = 5/40

          cos BCA = 1/8

          cos BCA = 0.125

             ÐBCA = arc cos 0,125

                         = 82,81^o

So, the measure of angle BCA is 82,81^o.