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.)