Cosine Formula

Cosine Formula

In trigonometry, the Cosine Rule states that the square of the length of any side of a given triangle is equal to the sum of the squares of the lengths of the other sides minus twice the product of the other two sides multiplied by the cosine of the angle contained between them. Cosine rule is also known as the law of cosines or Cosine Formula.

If a, b, and c are the lengths of the sides of a triangle ABC, then:

a² = b² + c² – 2bc cos α

b² = a² + c² – 2ac cos β

c² = a² + b² – 2ab cos γ

Derivation of Cosine

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

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

a² = b² + c² – 2bc cos α.

Hence, proved.

Solved Examples of Cosine x Formula

Example 1. Determine the angle of triangle ABC if AB = 21cm, BC = 14cm and AC = 7cm?

Solution:

As per the question we have following given data:

a = 21cm, b = 14cm and c = 7cm

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

− 2(14)(7)cos α =14² + 7² − 21²

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

cos α = 1

Thus, α = cos ⁻¹ 1 = 0°