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Sunday, November 11, 2012

Sum - Product, Two Angles Formula

Category: Trigonometry

"Published in Newark, California, USA"

On October 9, 2012, we derived the formulas for Sum and Difference of Two Angles Formula as follows:

Sin (θ + ϕ) = Sin θ Cos ϕ + Cos θ Sin ϕ          (equation 1)

Sin (θ - ϕ) = Sin θ Cos ϕ - Cos θ Sin ϕ            (equation 2)

Cos (θ + ϕ) = Cos θ Cos ϕ - Sin θ Sin ϕ          (equation 3)

Cos (θ - ϕ) = Cos θ Cos ϕ + Sin θ Sin ϕ          (equation 4)

If you add equation 1 and equation 2, the equation becomes

Sin (θ + ϕ) + Sin (θ - ϕ) = 2 Sin θ Cos ϕ

Sin θ Cos ϕ = ½ Sin (θ + ϕ) + ½ Sin (θ - ϕ)

If you subtract equation 1 and equation 2, the equation becomes

Sin (θ + ϕ) - Sin (θ - ϕ) = 2 Cos θ Sin ϕ

Cos θ Sin ϕ = ½ Sin (θ + ϕ) - ½ Sin (θ - ϕ)

If you add equation 3 and equation 4, the equation becomes

Cos (θ + ϕ) + Cos (θ - ϕ) = 2 Cos θ Cos ϕ

Cos θ Cos ϕ = ½ Cos (θ + ϕ) + ½ Cos (θ - ϕ)

If you subtract equation 3 and equation 4, the equation becomes

Cos (θ + ϕ) - Cos (θ - ϕ) = - 2 Sin θ Sin ϕ 

Sin θ Sin ϕ = - ½ Cos (θ + ϕ) + ½ Cos (θ - ϕ)

Sin θ Sin ϕ = ½ Cos (θ - ϕ) ½ Cos (θ + ϕ)

Therefore, the formulas for the transformation of Product to Sum of Two Angles are

Sin θ Cos ϕ = ½ Sin (θ + ϕ) + ½ Sin (θ - ϕ)

Cos θ Sin ϕ = ½ Sin (θ + ϕ) - ½ Sin (θ - ϕ)

Cos θ Cos ϕ = ½ Cos (θ + ϕ) + ½ Cos (θ - ϕ)

Sin θ Sin ϕ = ½ Cos (θ - ϕ) ½ Cos (θ + ϕ)

(Note: I strongly suggest that you must remember or memorize these formulas because you will use these formulas often in Integral Calculus especially the integration of product of two trigonometric functions with different angles.)

Consider the following formulas:

Sin (θ + ϕ) + Sin (θ - ϕ) = 2 Sin θ Cos ϕ

Sin (θ + ϕ) - Sin (θ - ϕ) = 2 Cos θ Sin ϕ

Cos (θ + ϕ) + Cos (θ - ϕ) = 2 Cos θ Cos ϕ

Cos (θ + ϕ) - Cos (θ - ϕ) = - 2 Sin θ Sin ϕ 

If x = θ + ϕ and y = θ - ϕ, the values of θ and ϕ are

           x = θ + ϕ                   x = θ + ϕ
           y = θ - ϕ                - (y = θ - ϕ)
        -----------------             -----------------
          x + y = 2 θ                 x - y = 2 ϕ
      
          θ ½ (x + y)             ϕ = ½ (x - y)

Substitute the values of θ + ϕ, θ - ϕ, θ, and ϕ to the four equations above, we have


Sin x + Sin y = 2 Sin ½ (x + y) Cos ½ (x - y)

Sin x - Sin y = 2 Cos ½ (x + y) Sin ½ (x - y)

Cos x + Cos y = 2 Cos ½ (x + y) Cos ½ (x - y)

Cos x - Cos y = - 2 Sin ½ (x + y) Sin ½ (x - y) 

  
Therefore, the formulas for the transformation of Sum to Product of Two Angles are


Sin x + Sin y = 2 Sin ½ (x + y) Cos ½ (x - y)

Sin x - Sin y = 2 Cos ½ (x + y) Sin ½ (x - y)

Cos x + Cos y = 2 Cos ½ (x + y) Cos ½ (x - y)

Cos x - Cos y = - 2 Sin ½ (x + y) Sin ½ (x - y) 

(Note: I strongly suggest that you must remember or memorize these formulas because you will use these formulas also for proving of trigonometric identities.)