Formulae to Transform Product into Sum or Difference

We have seen how to use formulae to expand or simplify trigonometric expressions. However, there are certain scenarios wherein we need to express the product of sine and cosine as a sum or a difference. We go about doing this by using the product to sum or sum to product formulae. 

The process of converting products into sums or vice versa is the thin line between finding an easy solution to a problem and being stuck without a solution. Hence, knowing a fixed set of formulae that will give us a simplified solution by converting functions into sums or differences goes a long way in solving for integrals in trigonometric functions. 

Product to sum formulas

There are a total of four product-to-sum formulae.  These are a staple in every product to sum scenario. We will discuss their derivations in a while. 

1. sinAcosB = [sin(A+B)+sin(A-B)]
2. cosAsinB = [sin(A+B)-sin(A-B)]
3. cosAcosB = [cos(A+B)-cos(A-B)]
4. sinAsinB = [cos(A-B)+cos(A+B)]

Derivation

Although we know there are four product to sum formulae, we need to understand how we derive these from trigonometric functions. For starters, let’s try to decode an example.

If we subtract sin(a – b) from sin(a + b)  from the functions sin(a – b) = sinacosb – cosasinb and sin(a + b) = sinacosb + cosasinb, we can derive a working product to sum formula for the product arising out of this solve which is cosacosb. 

 =+  

–

This should give us 

–

2

All we need to do now is multiply both sides with ½. We do this so we can isolate the sine and cosine products.

Which gives us

½(2

 .

Let’s try to understand this further with the help of an example. 

Let’s try to express the product as a sum that contains only sine or cosine. We know what the product to sum formula for sine and cosine is. All we need to do is expand the formula and substitute our known values and then simplify it.

The formula we will be using here is 1. sinAcosB =  [sin(A+B)+sin(A-B)].

Let’s substitute our known values.

= 1/2

What we did here was solve this for a sine function. Once we understand what the four product-to-sum formulas do, all we need to do is substitute and simplify.

Converting a cosine product into a sum

Let’s now use an example that will utilise a bit more in-depth equation to solve for cosine.

2

We know the cosine product to sum formula for cosine functions is. Let’s go ahead and use this to derive the sum for this equation. 

Let’s now substitute our known values into this formula. 

2

= [

=

Sum to product expressions

We now know how to express products as sums or differences. However, some situations require us to reverse the operation we followed, not which was the product to sum process. What we do here is we express sine or cosine sums as products using a method called the Sum to Product.

Let’s try to understand how this works. 

Let’s assume a simple example. Let’s say = a and  = b.

Therefore, a+b = +

=

= u

And a-b =  – 

=

= v.

Now we have some known values which we will be introducing into our known product to sum formula to derive a product from the sum of either a sine or a cosine function. 

As seen earlier, we will now substitute a and b with the derived values.

= 2

All other product to sum formulas are derived in a similar manner, by substituting derived values from the sums of a sine or cosine function.

Product to sum formulas

Just as is the case with product to sum formulas, we also have four product to sum formulas.

Let’s try to put this in use with the help of an example.

Because we have already seen how sum functions work, let’s see how a difference expression works.

Let’s try solving to derive the difference in the sine expression.

Let’s first identify the formula for the product to difference expression here. 

Substituting known values for a and b.

= 2

= 2

Using product to sum formulas, we can derive sums from sine and cosine products and vice versa. These formulas are particularly useful when a trigonometric function requires to be expressed as a sum or a difference, or when the product of such a trigonometric function requires to be expressed as a sum or difference. 

All sum or difference to product and product to sum or difference formulas revolve around sine and cosine functions. In case we do encounter functions other than sine or cosine, the easiest way of going about is first converting the said functions into sine or cosine and then proceeding with the substitution of known values.