Sum and difference of sines; sum and difference of cosines
These formulas convert sums and differences of sines and cosines into products: sin α + sin β = 2 sin((α + β) : 2) cos((α − β) : 2); sin α − sin β = 2 sin((α − β) : 2) cos((α + β) : 2); cos α + cos β = 2 cos((α + β) : 2) cos((α − β) : 2); cos α − cos β = −2 sin((α + β) : 2) sin((α − β) : 2). The first is proved this way: put x = (α + β) : 2, y = (α − β) : 2, so x + y = α, x − y = β, and sin(x + y) + sin(x − y) = 2 sin x cos y. The others are obtained similarly. The formulas help compute sums for angles not in the table (for example sin 75° + sin 15°), factor expressions, and find greatest and smallest values (sin α + cos α = √2 cos(α − π/4)). Be careful to compute the half-sum and half-difference correctly.
“Sum and product”: two rows of cards: sums such as sin 7α + sin 3α in one, products in the other; students find the correct pairs and check the half-sum and half-difference.