Cosine Rules

The cosine of an angle is a fundamental trigonometric ratio that relates the sides and angles of a right triangle. The cosine of an angle is defined as the ratio of the side adjacent to the reference angle and the length of the hypotenuse. Some certain laws or rules relating to the sides and angles of a triangle in terms of cosine trigonometric function. These rules are called the Cosine rule formula or Cosine law. If a, b and c are the sides of the triangle and A, B and C are the angles of the triangle. Then the cosine rule formula states that:

What is the Cosine Rule Formula?

Statement:

The cosine rule states that the square on any one side of a triangle is equal to the difference between the sum of the squares on the other two sides and twice the product of the other two sides and cosine of the angle opposite to the first side.

Explanation:

Consider the triangle shown in the above figure. In this triangle, ABC, a, b and c are the sides. According to the cosine rule proof, the square of the side BC i.e. ‘a’ is given as the product of the side ‘b’, side ‘c’ and the cosine of the angle A subtracted from the sum of the squares of the sides ‘b’ and ‘c’. It is Mathematically written as:

\[a^2 = b^2 + c^2 -2. Bc. Cos A\]

Laws of Cosine Trigonometric Function:

According to the laws of cosine, any one side of the triangle can be found when the other two sides and the angle opposite to the unknown side is given. The equations for the sides k, l and m of a triangle taken in order when ∠K, ∠L and ∠M are the angles opposite to the sides k, l, and m respectively  are given as:

\[k^2 = I^2 + m^2 -2. Im. Cos K\]

\[I^2 = m^2 + k^2 -2. mk. Cos L\]

\[m^2 = k^2 + I^2 - 2. kI. Cos M\]

Similarly, the cosine rule for angles of a triangle when all the three sides are given can be found as

Cos K = \[\frac {(I^2 + m^2 - k^2)}{2Im}\]

Cos L = \[\frac {(m^2 + k^2 - I^2)}{2mk}\]

Cos M = \[\frac {(k^2 + I^2 - m^2)}{2kI}\]

Cosine Rule Proof:

Data: 

Let us consider the triangle ABC as shown in the figure below. The sides of the triangle AB, BC and AC measure ‘c’, ‘a’ and ‘b’ units respectively. 

To Prove: 

\[c^2 = a^2 + b^2 - 2. ab. Cos C\]

Construction:

Draw BD perpendicular to AC at the point D

Cosine Rule Proof:

Cosine Rule Examples:

1. In the triangle shown below, find the length of c.

Solution:

Given data: a = 8 cm, b = 11 cm and ∠C = 370

Cosine rule states that   

\[c^2 = a^2 + b^2 - 2ab . Cos C \]   

\[c^2 = 8^2 + 11^2 - 2 \times 8 \times 11 Cos 37^o \]

\[c^2 = 64 + 121 - 176 \times 0.8 \]

\[c^2 = 185 - 140.8 \]

\[c^2 = 44.2 \]

\[c^2 =  ± \sqrt {44.2}\]

\[c^2 =  ± 6.65 \]

However, length cannot have a negative value. So, c = 6.65 units.

Fun Facts:

• Though the concept of cosine rule did not exist in the 3rd century BC, there is evidence of concepts similar to cosine rule examples in the Mathematical works of Euclid, the father of geometry.

• Laws of cosine are used in solving triangles and circles by a Mathematical procedure called triangulation.

Why Should You Learn the Cosine Rules? 

Learning the cosine rules is essential for all mathematics students. Here are some reasons as to why you should learn the concept of cosine rules: 

• Cosine rules allow you to find the sides and angles of a triangle. You can use these rules to determine a side of the triangle when two of them are known. Similarly, you can find every angle of the triangle when all the sides are known. 

• The cosine rules give you a better understanding of the trigonometric functions and enhance your knowledge of the concept. 

• Since trigonometry is a crucial topic of mathematics, you should know everything about the rules of cosine. It will aid you in scoring the highest marks in your tests and final exams.

Proving the Cosine Rules 

There are many methods to prove the rules of cosine. Below are the formulas you can use to prove the law of cosines: 

• Trigonometry

• Distance Formula 

• Pythagorean Theorem 

• Ptolemy’s theorem

• Comparing areas 

• Geometry of circles 

• Using law of sines 

• Vectors 

Out of the above methods, the Pythagorean theorem is the most used way of proving the laws of cosine. However, you can use any of the above methods in your exam to provide proof of cosine rules.

How to Study the Cosine Rules?

Studying trigonometric functions will be much easier if you have the right study material for it. You can learn the cosine rules using the resources provided on Vedantu’s online learning platform for absolutely free. These study materials will allow you to understand how to prove cosine rules, formulas of cosine, and how to solve questions using the law of cosines. Here are some tips and tricks you can use to start learning the cosine rules:   

• Once you have studied the concept of cosine rules from the textbook, try to solve the exercise questions based on these rules to test your knowledge. 

• Since cosine rules is an important concept, studying from the textbook only will not be enough to learn the topic. You should use different reference books to practice the questions related to the law of cosines and enhance your mathematical skills. 

• Go through as many examples and illustrations as you can to have a better understanding of the cosine rules. 

• Use the solved question papers to understand how to solve different types of questions based on the cosine rules. 

• Once you have completed the textbook and reference book questions, you can move to the previous year question papers to understand what type of questions come in your final exam. 

• Use Vedantu’s revision notes, important questions, NCERT solutions, and other study materials to get a clear idea of cosine rules. 

• Try as many questions as you can to improve your problem-solving skills and become more proficient in solving questions based on the law of cosines.