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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 interrelationship of these elements using the cosine function.
1.  What is Law of Cosines? 
2.  Law of Cosines Formula 
3.  Proof of Law of Cosines 
4.  Finding Missing Sides and Angles 
5.  FAQs on Law of Cosines 
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: a^{2} = b^{2} + c^{2} − 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:
 find a when b, c, and A are given (or)
 find b when a, c, and B are given (or)
 find c when a, b, and C are given (or)
 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.
a^{2} = b^{2} + c^{2 } 2bc·cosA
b^{2} = c^{2} + a^{2}  2ca·cosB
c^{2} = a^{2} + b^{2}  2ab·cosC
Proof of Law of Cosines
There is more than one way to prove the law of cosine. Let's prove it using trigonometry. Consider the following figure.
In ΔABM we have,
sin A = BM/AB = h/c
and,
cos A = AM/AB = r/c
From equation (1) and (2), we get h = c(sin A) and r = c(cos A)
By Pythagoras Theorem in ΔBMC,
a^{2 }= h^{2} + (b  r)^{2}
Substitute h = c(sin A) and r = c(cos A) in equation (3).
a^{2} = (c(sinA))^{2} + (b  c(cosA))^{2}
= c^{2}sin^{2}A + b^{2} + c^{2}cos^{2}A  2bc·cosA
= c^{2}(sin^{2}A + cos^{2}A) + b^{2}  2bc·cosA
= c^{2} + b^{2}  2bc·cosA
Hence, proved.
Finding Missing Length and Angles Using the Law of Cosines
As discussed above, law of cosines can be used to calculate the missing parameters of a triangle, given the required known elements. Let us have a look at the following steps to understand the process of finding the missing side or angle of a triangle using the cosine law.
 Step 1: Note down the given data(side lengths and measure of angles) for the triangle and identify the element to be calculated.
 Step 2: Apply the cosine rule formulas,
a^{2} = b^{2} + c^{2 } 2bc·cosA
b^{2} = c^{2} + a^{2}  2ca·cosB
c^{2} = a^{2} + b^{2}  2ab·cosC
where, A, B, and C are the vertices of a triangle, and their opposite sides are represented as a, b, and c respectively.  Step 3: Express the obtained result with suitable units.
Let us consider a few examples to find the missing side and angle of a triangle.
Example: Look at the figure shown below.
We need to find the measure of ∠A.
We will use the formula
a^{2} = b^{2} + c^{2}  2bc.cosA. Substitute 10 for 'a', 7 for 'b' and 5 for 'c'.
10^{2 }= (7)^{2} + (5)^{2}  2(5)(7)·cosA
70·cos A = 26
cos A = 13/35
A = 111.8º
In this example, we used the law of the cosine equation to find the missing angle. Now, let us use the law of the cosine equation to find the missing side.
Example: Two sides of a triangle measure 72 in and 50 in with the angle between them measuring 49º let us find the missing side.
Solution:
Substitute 72 for b, 50 for c and 49º for A.
Using the law of cosines formula,
a^{2 }= b^{2} + c^{2}  2bc·cosA
a^{2} = (72)^{2} + (50)^{2}  2(72)(50)cos49º
a^{2} = 5184 + 2500  (7200)(0.656)
a^{2} = 5184 + 2500  4723.2
a^{2} = 2960.8
a ≈ 54.4
So, the missing length of the side is 54.4 inches.
Important Notes on Law of Cosines:
 Three different versions of the law of cosine are:
a^{2} = b^{2} + c^{2 } 2bc·cosA
b^{2} = c^{2} + a^{2}  2ca·cosB
c^{2} = a^{2} + b^{2}  2ab·cosC  Pythagoras Theorem is a generalization of the Law of Cosine.
 The law of cosine can be applied in any triangle.
Challenging Question: A spider is lost in its web. Look at the figure below. Can you find the value of x?
Examples Using Law of Cosines

Example 1: A boy is standing at point A and two boats are located at points, B and C, such that the positions of all three form a triangle. If the measure of angle A is 36º with the lengths AB and AC measuring 2.5 ft and 1.8 ft respectively, determine the distance between the two boats floating at the lake.
Solution:
The law of cosine is expressed in three different ways.
We will use the formula a^{2} = b^{2} + c^{2}  2bc·cos A, because the required side is opposite to ∠A.
Substitute 1.8 for 'b', 2.5 for 'c' and 36º for angle A.
a^{2} = b^{2} + c^{2}  2bc·cosA
a^{2} = (1.8)^{2} + (2.5)^{2}  2(1.8)(2.5)cos 36º
a^{2} = 3.24 + 6.25  (9)(0.8)
a^{2} = 3.24 + 6.25  7.2
a^{2} = 2.29
a ≈ 1.5
Answer: The length across the boats is 1.5 feet.

Example 2: A farmer has a huge field in the shape of a triangle. The two sides of the field measure 624 ft and 327 ft and the angle between them measures 93º. Calculate how much fencing is needed to enclose the field?
Solution:
We will use the definition of law of cosine: a^{2} = b^{2} + c^{2}  2bc·cosA.
Substitute 624 for 'b', 327 for 'c' and 93º for A.
a^{2} = b^{2} + c^{2}  2bc·cosA
a^{2} = (624)^{2} + (327)^{2}  2(624)(327)cos 93º
a^{2} = 389376 + 106929  (408096)(0.05)
a^{2} = 389376 + 106929 + 20404.8
a^{2} = 516709.8
a ≈ 719
So, the perimeter of the triangular field is 624 + 327 + 719 = 1670ft
Answer: The farmer will need 1670 ft of fencing.

Example 3: The adjacent sides of a parallelogram measure 6 in and 10 in with the angle between them measuring 79º. Can you determine the length of the diagonal of the parallelogram?
Solution:
Given: Let ABCD be a parallelogram, such that, CD = 6 in, BC = 10 in.
We know that the adjacent angles of a parallelogram are supplementary.
So, ∠B = 180º  ∠C = 180º  79º = 101º.
Also, the opposite sides of a parallelogram are equal.
So, AB = CD = 6 in.
Let's apply the cosine rule in ΔABC.
Substitute 10 for 'a', 6 for 'c' and 101º for B.
b^{2} = a^{2} + c^{2}  2ac·cosB
b^{2} = (10)^{2} + (6)^{2}  2(10)(6)cos101º
b^{2} = 100 + 36  (120)(0.19)
b^{2} = 100 + 36 + 22.8
b^{2} = 158.8
b ≈ 12.6
Answer: The length of the diagonal is approximately 12.6 inches.
FAQs on Law of Cosines
What is Law of Cosines in Trigonometry?
The law of cosines is used to find the missing side of a triangle when its two sides and the included angle is given. There are three laws of cosines and we choose one of them to solve our problems depending on the available data.
 a^{2} = b^{2} + c^{2 } 2bc·cosA
 b^{2} = c^{2} + a^{2}  2ca·cosB
 c^{2} = a^{2} + b^{2}  2ab·cosC
where, A, B, and C are the vertices of a triangle, and their opposite sides are represented by the small letters a, b, and c respectively.
What is Law of Cosines Used For?
The Law of Cosines can be used to find the unknown parts of an oblique triangle(nonright triangle), such that either the lengths of two sides and the measure of the included angle is known (SAS) or the lengths of the three sides (SSS) are given.
What are the Possible Criteria for Law of Cosines?
In order to use the law of cosines, either two sides of the triangle and the measure of included angle(SAS) or the length of all three sides of the triangle(SSS) should be known.
What is the Formula for Law of Cosines?
The formula for law of cosines is given as,
 a^{2} = b^{2} + c^{2 } 2bc·cosA
 b^{2} = c^{2} + a^{2}  2ca·cosB
 c^{2} = a^{2} + b^{2}  2ab·cosC
where, A, B, and C are the vertices of a triangle, and their opposite sides are represented as a, b, and c respectively.
How to Derive Law of Cosines Formula?
There is more than one way to derive the law of the cosine formula. A few of them are given as,
 Using basic concepts of trigonometry.
 Using vector algebra.
 Using the law of sine.
 Using the coordinate geometry to find the distance between two coordinate points.
What is the Application of Law of Cosines Formula?
The law of cosines formula finds application in finding 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.
How to Find the Missing Side or Angle of Triangle Using Law of Cosines?
Law of cosines can be used to find the missing side or angle of a triangle by applying any of the following formulas,
 a^{2} = b^{2} + c^{2 } 2bc·cosA
 b^{2} = c^{2} + a^{2}  2ca·cosB
 c^{2} = a^{2} + b^{2}  2ab·cosC
Here, A, B, and C are the vertices of a triangle, and their opposite sides are represented using letters a, b, and c respectively.
Does the Law of Cosine Work in All Triangles?
Yes, the law of cosines can be applied to all the triangles. It stands true for both right and oblique triangles.
Who Invented the Law of Cosines?
The elements by Euclid contributed to the discovery of the law of cosines. Jamshīd alKāshī, a Persian mathematician, was the first to provide the first explicit statement of the law of cosines in the 15th century.
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