Lesson 12 of 13 · 8 min
Sum and product formulas
NCERT §3.4
Two cabins, at angles C and D, are loaded at the same time. The midpoint of the chord between them sits on the spoke exactly half-way between them. That one picture turns sums of sines and cosines into products.
The lesson in notes
In short
Adding and subtracting the sum and difference formulas gives products as sums: 2 cos A cos B = cos (A + B) + cos (A − B) and −2 sin A sin B = cos (A + B) − cos (A − B).
In the same way, 2 sin A cos B = sin (A + B) + sin (A − B) and 2 cos A sin B = sin (A + B) − sin (A − B).
Reading these backwards turns sums into products: cos A + cos B = 2 cos ((A + B)/2) cos ((A − B)/2) and cos A − cos B = −2 sin ((A + B)/2) sin ((A − B)/2).
Likewise, sin A + sin B = 2 sin ((A + B)/2) cos ((A − B)/2) and sin A − sin B = 2 cos ((A + B)/2) sin ((A − B)/2).
Worked values: sin 75° + sin 15° = 2 sin 45° cos 30° = 2 × (1/√2) × (√3/2) = √6/2, and cos 75° − cos 15° = −2 sin 45° sin 30° = −1/√2.
Products turn into sums just as easily: 2 sin 75° cos 15° = sin 90° + sin 60° = 1 + √3/2.
These formulas are the standard tool for simplifying quotients: (sin 7x + sin x)/(cos 7x + cos x) = (2 sin 4x cos 3x)/(2 cos 4x cos 3x) = tan 4x, wherever cos 4x cos 3x ≠ 0.