Working with Trigonometric Ratios on the Coordinate Plane
How to Prove an Equality by Using Periodicity Identities
How to Combine Reference Angles with Other Techniques to Solve Trigonometric Equations

How to Express Products of Trigonometric Functions as Sums or Differences

If you can break up a product of trig functions into the sum of two different terms, each with its own trig function, doing the math becomes much easier. In pre-calculus, problems of this type usually say "express the product as a sum or difference." In the following example, you'll make the conversion from a product to a sum.

You have three product-to-sum formulas to digest: sine multiplied by cosine, cosine multiplied by cosine, and sine multiplied by sine.


Suppose that you're asked to express 6 cos q sin 2q as a sum. Rewrite this expression as 6 sin 2q cos q (thanks to the commutative property) and then plug what you know into the formula to get


For example, to express


as a sum, rewrite it as the following:


To express sin 5x cos 4x as a sum, rewrite it as the following:

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