Transformation formulae

Formulae:

1. Sin (A + B) + sin (A - B) = 2 sinA cosB
2. sin (A + B) - sin (A - B) = 2 cosA sinB
3. cos (A + B) + cos (A - B) = 2 cosA cosB
4. cos (A - B) - cos (A + B) = 2 sinA sinB
5. sinC + sinD = 2 sin (c + D)/2 cos (C -D)2
6. cosC + cosD = 2 cos (c + D)/2 cos (C -D)2
7. cosC - cosD = -2 sin (c + D)/2 sin (C -D)2

Ex 1: If A + B + C = π then sin 2A + sin 2B + sin 2C =?

Solution:

sin 2A + sin 2B + sin 2C

= 2 sin(2A + 2B)/2 cos(2A – 2B)/2 + sin 2C

= 2 sin (A + B) cos (A - B) + sin 2C

= 2 sin C [cos (A - B) + cos C]

= 2 sin C [cos (A - B) - cos (A + B)]

= 4 sinA sinB sinC

Ex 2:Find the value of cos 15

Solution:

We have cos 2θ = 2 cos² θ - 1

=> cos 30 = 2 cos² 15 - 1

=> √3/2 + 1 = 2 cos² 15

=> (√3 + 2)/ 4 = cos² 15

=> (2√3 + 4)/ 8 = cos² 15

=> (√3 + 1)² / 8 = cos² 15

=> cos 15 = (√3 + 1)/2√2

Ex 3: If A + B + C = π then Tan A/2 Tan B/2 + Tan B/2 Tan C/2 + Tan C/2 Tan A/2 =?

Solution:

A + B + C = π

=> A/2 + B/2 = π/2 – C/2

=> Tan (A/2 + B/2) = Tan (π/2 – C/2)

=> ([Tan A/2 Tan B/2]/1 - Tan B/2 Tan A/2) = cot C/2

=> ([Tan A/2 Tan B/2]/1 - Tan A/2 Tan B/2) = 1/Tan C/2

=> Tan A/2 Tan C/2 + Tan B/2 Tan C/2 = 1 - Tan A/2 Tan B/2

=> Tan A/2 Tan B/2 + Tan B/2 Tan C/2 + Tan C/2 Tan A/2 = 1