< Trigonometry < For Enthusiasts
In this section, we shall see how to convert a product of two trigonometric functions into a sum or difference of two such functions, and vice versa.
Product into sum
We have already seen that
and
Adding these two equations and dividing both sides by 2, we get
Subtracting the second from the first equation and dividing both sides by 2, we get
We also know that
and
Adding these two equations and dividing both sides by 2, we get
Subtracting the first from the second equation and dividing both sides by 2, we get
Thus we can express:
- The product of a sine and cosine as the sum or difference of two sines;
- The product of two cosines as the sum of two cosines;
- The product of two sines as the difference of two sines.
Sum into product
Let and . Then
Substituting into the above expressions and multiplying both sides by two in each of them, we have:
Note the negative sign in the last formula.
These formulae are sometimes expressed in words, e.g.
- cos plus cos = two cos half sum cos half diff.
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