Sum to Product Formulae

And now the Sum to Product Formulae:

sin x + sin y = 2sin((x + y)/2) cos((x - y)/2)

sin x - sin y = 2cos((x + y)/2) sin((x - y)/2)

cos x + cos y = 2cos((x + y)/2) cos((x - y)/2)

cos x - cos y = -2sin((x + y)/2) sin((x - y)/2)

Example

This little example will not only grant thou some practice, but it will also prove the formulae!

Express cos47° + cos59° as a product.

Solution

Remember that the sum of two cosines formula is

cos(A + B) + cos(A - B) = 2cosAcosB

Now, Trig Frog says that A + B = 47° and A - B = 59°. So, thou shouldst solve the resulting system of equations:

A + B = 47°

A - B = 59°

Adding these two equations results in:

2A = 106° Divide by 2.

A = 53°

Subtracting the two equations results in:

2B = -12° Divide by 2.

B = -6°

So:

cos47° + cos59° = 2cos53°cos(-6°) Remember that cosine is an even function.

Answer: 2cos53°cos6°

"How does this prove the formulae?" thou might ask. Well, the procedure can be done in general:

A + B = some angle x

A - B = some angle y

The solutions are A = 1/2(x + y) and B = 1/2(x - y). If these values are substituted into the Product-to-Sum Formulae, the result is an set of formulae that express a sum as a product...the Sum-to-Product Formulae!