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.

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.