Fundamental law of trigonometry Formula

Fundamental law of trigonometry

Fundamental law of trigonometry Formula: Let Alpha(α) and beta (β) any two angle (real number),


cos (α – β) = cos α cos β + sin α sin β Which is called Fundamental law of trigonometry Formula.

Fundamental law of trigonometry


we sages that Alpha(α)  greater than beta (β)  and beta greater than zero α > β> 0 

Consider a unit circle with center at origin

Let Terminal sides of angle α and β cut the unit circle at A and B respectively

First terminal ray Angle ∠ AOB = α – β


Take a point C on the unit circle

so that

  Angle ∠ X O C = Angle ∠ AOB = α – β

join A, B and C, D

Now angles Alpha(α) and beta (β) and α – β are in the standards position.


the coordinate of A arc (cos α, sin α)

The coordinate of Arc B (cos β, sin β)

The coordinate of Arc C are  (cos α – β, sin α – β)


The coordinate of Arc D (1, 0)

Now Δ AOB and Δ COD are congruent.


|AB| = |CD|   this implies that   |AB|² = |CD|²

Using the distance formula 

(Cos α – Cos β)²+(Sin α – Sin β)²= [Cos (α – β)-1]²+[Sin(α – β) – 0]²

this implies that

Cos²α + Cos²β-2Cos α  Cos β + Sin² α Sin² β – 2 Sin α sin β

this is equal to 

Cos² (α – β) +1 -2 Cos (α -β) + Sin²(α −β)

This is equal to 

2 – 2 (Cos α Cos β+ Sin α Sin β) = 2 – 2 Cos (α – β)

After cancelation, we get the result

cos (α – β) = cos α cos β + sin α sin β

When Alpha(α) greater than  beta (β) greater than zero then this formula is true for all values of Alpha(α) and beta (β)

α >β> 0