Cosine Rule

Cosine Rule commonly referred to as the Law of Cosines in Trigonometry establishes a mathematical connection involving all three sides of a triangle and one of its angles. Cosine Rule is most useful for solving the unknown information of a triangle. For example, when all three sides of a triangle are known, the Cosine Rule allows the determining of any angle measurement. Similarly, if two sides and the included angle between them are known, this rule facilitates the calculation of the third side length.

The Cosine Rule is a relationship between the lengths of a triangle’s sides and the cosine of one of its angles, allowing us to calculate distances and angles. When computing the third side of a triangle if two sides and their included angle are given, and when computing the angles of a triangle if all three sides are known, in that case, the Cosine Rule plays a valuable role.

In this article, we will discuss the introduction, definition, properties, formula of the Cosine Rule, and its meaning. We will also understand the proof of the Cosine Rule. We will also solve various examples and provide practice questions based on Cosine Rule for a better understanding of the concept of this article.

What is the Cosine Rule?

Cosine Rule, also known as the Law of Cosines, establishes a mathematical relationship between the lengths of a triangle’s sides and the cosine of one of its angles. This formula is used as an instrumental tool in calculating side lengths or angle measures within a triangle.

Cosine Rule can be applied when either all three sides of the triangle are known or when information about two sides and the included angle is provided. The Cosine Rule allows us to calculate the length of a specific side of a triangle when we have information about the other two sides and the angle of the triangle.

Definition of Cosine Rule

The Cosine Rule is defined as “the square of one side of a triangle, equals the sum of the squares of the other two sides, subtracted by twice the product of those two sides and the cosine of the angle between them.”

This fundamental formula helps to establish a connection between the sides and angles within a triangle. It is useful to find the unknown information in any triangle. For example, if the lengths of two sides of a triangle and the included angle are known, this rule helps the calculation of the third side of the triangle.

Cosine Rule Formula

Cosines Rule Formula is applicable to determine the unknown side of a triangle when the lengths of two sides and the angle between them are given. This scenario is used in the case of a SAS triangle, where Side-Angle-Side information is available. If we denote the angle of a triangle as α, β, and γ, then their respective opposite sides are denoted by the lowercase letters a, b, and c. The Cosines Rule formula is used to:

• find a when b, c, and α are given; b when a, c, and β are given; c when a, b, and γ are given.
• find any angle of the triangle when their respective opposite sides a, b, and c are given.

There are three variations of the cosine law, and we choose one of them to use in problem-solving depending on the given data.

a² = b² + c² – 2bc cos α

b² = a² + c² – 2ac cos β

c² = a² + b² – 2ab cos γ

Cosine Rule of Triangle

Let us consider a triangle ABC with side lengths represented by a, b, and c.

The Cosine Rule formula can be expressed as follows:

a² = b² + c² – 2bc cos A

b² = a² + c² – 2ac cos B

c² = a² + b² – 2ab cos C

Where,

• A, B, and C are the angles of triangle, and
• a, b, and c are the sides of triangle.

Derivation of the Cosine Rule

Cosine Rule can be derived

• Using Geometry
• Using Algebraic Manipulation

Let’s discuss these in detail as follows:

Derivation Using Geometry

Cosine Rule can be demonstrated through various methods. In this case, we’ll opt for a trigonometric approach to prove it. Stating the law of cosines, it applies to a triangle ABC, where the sides are denoted as a, b, and c.

c² = a² + b² – 2ab cosγ

Now let us prove this law.

Let us suppose a triangle ABC. From the vertex of angle B, we draw a perpendicular touching the side AC at point D. “h” is the height of the triangle.

Now in triangle BCD, as per the trigonometry ratio, we know;

cos γ = CD/a (cos θ = Base/Hypotenuse)

⇒ CD = a cos γ ………… (1)

Put equation (1) from side b on both the sides, we get;

b – CD = b – a cos γ

⇒ DA = b – a cos γ

Again, in triangle BCD, as per the trigonometry ratio,

sin γ = BD/a (sin θ = Perpendicular/Hypotenuse)

⇒ BD = a sin γ ……….(2)

Now using Pythagoras theorem in triangle ADB, we get;

c² = BD² + DA² (Hypotenuse² = Perpendicular² + Base² )

Put the value of DA and BD from equation 1 and 2, we get;

c² = (a sin γ )² + (b – a cos γ)²

⇒ c² = a² sin² γ + b² – 2ab cos γ + a² cos² γ

⇒ c² = a² (sin² γ + cos²γ) + b² – 2ab cos γ

By trigonometric identities, we know;

sin²θ+ cos²θ = 1

⇒ c² = a² + b² – 2ab cos γ

Hence, proved.

Derivation Using Algebra

Let us consider a triangle with sides a, b, and c and their respective angles by α, β, and γ.

We know, from the law of sines,

a/sin α = b/sin β = c/sin γ

The sum of angles inside a triangle is equal to 180 degrees

Therefore, α+β+γ = π

Using the third equation system, we get

c/sin γ = b/sin (α + γ) ———– (1)

⇒ c/sinγ = a/sin α

Using angle sum and difference identities, we get,

sin (α + γ) = sin α cos γ + sin γ cosα

⇒ c (sin α cos γ + sin γ cos α ) = b sin γ

⇒ c sin α = a sin γ

Dividing the whole equation by cos γ,

c (sin α + tan γ cos α) = b tan γ

⇒ c sin α /cos γ = a tan γ

⇒ c²sin² α / cos² γ = tan γ

From equation 1, we get,

c sin α / b – c cos α = tan γ

⇒ 1 + tan² γ = 1/cos² γ

⇒ c² sin² α (1+ (c² sin²α / (b – c cos α )²)) = a² (c² sin²α / (b – c cos α )²)

Multiplying the equation by (b – c cos α )² and arranging it,

a² = b² + c² – 2bc cos α.

Hence, using algebraic manipulation cosine rule is proved.

Properties of Cosine Rule

The properties of cosine rule are listed below

• To determine the side lengths of triangle ABC, we can express it as a² = b² + c² – 2bc cos α, b² = a² + c² – 2ac cos β and c² = b² + a² – 2ba cos γ.
• To calculate the angles of triangle ABC, the Cosine Rule is expressed in the following manner: cos α = [b² + c² – a²]/2bc, cos β = [a² + c² – b²]/2ac and cos γ = [b² + a² – c²]/2ab.
• Sine Rule is expressed as a/sin α = b/sin β = c/sin γ.
• Cosine Rule is also known as the Law of Cosines.
• Cosine Rule is a formula which helps to calculate the sides and angles of a triangle.
• Pythagoras Theorem is an application of the Cosine Rule which only holds true for right-angle triangles.
• The Cosine Rule can be applied or utilized in any triangle.

Where to use Cosine Rule?

The Cosine Rule is useful for finding:

• The third side of a triangle by using the lengths of two known sides and the angle formed between them.
• The angles of a triangle by using all three sides of the triangle.

Examples of the Cosine Rule

We can use cosine rule for:

• Finding Sides
• Finding Angles

Let’s discuss the method for finding sides and angles in detail as follows:

Finding Missing Length Using the Cosine Rule

Cosine Rule can be used to calculate the unknown parameters of a triangle when all known elements are given. Let us understand the process of finding out the missing side or angle of a triangle using the Cosine Rule.

Step 1: Note down the given parameters like side lengths and measure of angles for the triangle and identify the element to be calculated.

Step 2: Apply the cosine rule formulas,

• a² = b² + c² – 2bc cos α
• b² = a² + c² – 2ac cos β
• c² = a² + b² – 2ab cos γ

where, α, β, and γ are the angle of a triangle, and their opposite sides are represented as a, b, and c respectively.

Step 3: Represent the result with suitable units.

Cosine Rule To Find Angles

Cosine Rule can be used to find unknown angles in a Triangle using the formula given below:

• cos A = (b² + c² − a²)/2bc
• cos B = (a² + c² − b²)/2ac
• cos C = (a² + b² − c²)/2ab

Example 4 in Solved Examples deals with Cosine Rule to Find Angles

Sine and Cosine Rule

Sine Rule and Cosine Rule are important rules in Trigonometry to establish the relation between angles and sides of a triangle. A detailed comparioson between Sin Rule and Cosine Rule is discussed in the table below:

Also, Check

• Trigonometry
• Trigonometric Formulas
• Cosine Formulas

Solved Examples on Cosine Rule

Example 1. Determine the angle of triangle ABC if AB = 42cm, BC = 37cm and AC = 26cm?

Solution:

As per the question we have following given data:

a = 42cm b = 37cm and c = 26cm

Formula of Cosine Rule: a² = b² + c² − 2bc cos α

So, 42² = 37² + 26² − 2(37)(26) cos α

cos α = 37² + 26² − 42² /(2)(37)(26) 985

After solving the cos α we get the value of α as

cos α = 1071/2184

⇒ cos α = 0.4904

Thus, α = cos ⁻¹ 0.4904 = 60.63°

Example 2. Two sides of a triangle measure 70 in and 50 in with the angle between them measuring 49º. Find the missing side.

Solution:

Put the value 72 inch for b, 50inch for c and 49º for α.

Using the Cosine Rule formula,

a² = b² + c² – 2bccos α

⇒ a² = (70)² + (50)² – 2(72)(50)cos49º

⇒ a² = 4900 + 2500 – (7200)(0.656)

⇒ a² = 4900+ 2500 – 4723.2

⇒ a² = 2676.8

⇒ a ≈ 51.73

So, the missing length of the side is 51.73 inches.

Example 3. How long is side “c”, when we know the angle C = 37°, and sides a = 9 and b = 11.

Solution:

The Cosine Rule is c² = a² + b² − 2ab cos(C)

Put the given values we know: c² = 9² + 11² − 2 × 9 × 11 × cos(37º)

c² = 81 + 121 − 198 × 0.798

⇒ c² = 43.99

⇒ c = √43.99 = 6.63.

Example 4: Find Angle “C” using the Cosines Rule (angle version). If in this triangle we know the three sides: a = 8, b = 6 and c = 7.

Solution:

Use The Cosines Rule to find angle C :

cos C= (a² + b² − c²)/2ab

⇒ cos C = (8² + 6² − 7²)/2×8×6

⇒ cos C = (64 + 36 − 49)/96

⇒ cos C = 51/96

⇒ cos C = 0.53125

⇒ C= cos⁻¹(0.53125)

⇒ cos C = 57.9°

Practice Question on Cosine Rule

Q1. Determine the angle of triangle ABC if AB = 32cm, BC = 30cm and AC = 24cm.

Q2. Two sides of a triangle measure 62 in and 40 in with the angle between them measuring 50º. Find the missing side.

Q3. How long is side “c”, when we know the angle C = 47º, and sides a = 11 and b = 15.

Q4. Find Angle “C” using the Cosines Rule (angle version). If in this triangle we know the three sides: a = 9, b = 6 and c = 8.

Q5. Express sin 12θ + sin 4θ as the product of sines and cosines.

Cosine Rule – FAQs

What is the Cosine Rule for a Triangle?

According to the Cosine Rule, the square of one side of a triangle, equals the sum of the squares of the other two sides, subtracted by twice the product of those two sides and the cosine of the angle between them. Mathematically it is written as: a² = b² + c² – 2bc cos γ .

When can we use the Cosine Rule?

The Cosine Rule is useful for finding: The third side of a triangle by using the lengths of two known sides and the angle formed between them. The angles of a triangle by using all three sides of triangle.

How to Find Angles using the Cosine Rule, when Three Sides of Triangle are given?

If γ is the included angle, then

cos γ = (a² + b² – c²)/2ab

Where a,b and c are the sides of the triangle.

What is the Cosine Formula?

The Cosine formula to find the side of the triangle is given by:

c = √[a² + b² – 2ab cos γ ] where a,b and c are the sides of the triangle.

What is the Sine Rule formula?

As per the sine rule, if α, β , and γ are the length of sides of a triangle and A, B, and C are the angles, then a/sin α = b/sin β = c/sin γ

Does Pythagoras’ Theorem only Work for Cosine?

Pythagoras’ theorem tells us that the square of the hypotenuse side is equal to the sum of squares of the other two sides. However, it works for right-angled triangles so that we can check whether a triangle has a right angle or not.

How to Find the Missing Side or Angle of Triangle Using Cosine Rule?

• To determine the side lengths of triangle ABC, we can express it as: a² = b² + c² – 2bc cos α, b² = a² + c² – 2ac cos β and c² = b² + a² – 2ba cos γ
• To calculate the angles of triangle ABC, the cosine rule express in the following manner: cos α = [b² + c² – a²]/2bc, cos β = [a² + c² – b2]/2ac and cos γ = [b² + a² – c²]/2ab

Does the Cosine Rule Work in All Triangles?

Yes, the Cosine Rule can be applied to all the triangles. It stands true for both right and oblique triangles.

What is the difference between Cosine and Sine?

In a right angled triangle Cosine is the ratio of its base to hypotenuse whereas the Sine is the ratio of perpendicular and hypotenuse.

Why is Cosine Rule important?

• It helps in derivation of Pythagoras theorem and triangle inequality.
• It is used to calculate the unknown length of a side and angle of triangle.

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