Click here to see part one of this four week series.
Click here to see part two of this four week series.
Click here to see part three of this four week series.
Now that we have discovered some useful trigonometric identities, we can continue to build on them and create many more. There are an infinite number of trigonometric identities out there (not all of them have been created of course), but we will stick to two in this post: the product-to-sum formulas and the sum-to-product formulas.
Take the four angle addition/subtraction formulas we discovered in our first week. I will use A and B as our letters rather than alpha and beta.
1. sin(A + B) = sinAcosB + cosAsinB
2. sin(A – B) = sinAcosB – cosAsinB
3. cos(A + B) = cosAcosB – sinAsinB
4. cos(A – B) = cosAcosB + sinAsinB
These can be messed with very easily to create some new formulas. For instance, adding together the first two formulas gives:
sin(A + B) + sin(A – B) = sinAcosB + cosAsinB + sinAcosB – cosAsinB
2sinAcosB = sin(A + B) + sin(A – B)
sinAcosB = [sin(A + B) + sin(A – B)]/2
We now have a new identity. This can be now be used to solve a whole new range of problems and generate a whole new range of identities. The same steps can be done by subtracting the second equation from the first, adding the third and fourth together, and subtracting the fourth from the third. This creates the four product-to-sum identities.
These four formulas can be rewritten in a way that converts the sum into a product. Let's rewrite the variables as the following:
a + b = A
a – b = B
Making this change, we can then perform some operations to get a whole new set of formulas. These are called the sum-to-product identities.
These can then be built upon to generate whole new sets of formulas as well. Though the actual mathematics here might be a bit complicated, the idea is simple. Mathematics is always continuing to be developed, and this can be done through building upon previous ideas to form new ideas that help solve new problems. Trigonometry is a great place to see this sort of thing happen.