Sum-to-Product Identities: Uses & Applications

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  • 0:01 Sum-To-Product Identities
  • 1:44 Uses and Applications
  • 1:58 Example 1
  • 2:50 Example 2
  • 4:00 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this video lesson, you will learn about the four sum-to-product identities for the sine and cosine functions. Learn how to go from the sum or subtraction of two functions to the product of two functions along with the two angles.

Sum-to-Product Identities

The sum-to-product identities are the true trigonometry statements that tell you how to turn the sum or subtraction of two trig functions into the product of two trig functions. Think of these definitions as telling you what something is equal to. You can go back and forth between the definition and the term that is being defined. We call them identities, plural, because we have more than one. We have four of them! All of these identities deal exclusively with just the cosine and sine functions. Let's take a look at what they look like.

Sum-to-Product Identities
sum identities

It's a whole lot of sines and cosines! But notice some of the patterns. Seeing the patterns will help you to remember these formulas. Notice that when you have the sum or subtraction of two sine functions that you end up with the product of the sine and cosine functions. The argument of the first function is x + y over 2, and the argument of the second function is x - y over 2.

If we have the sum of two sine functions, then our first function is a 2. If we have subtraction, then our first function is the cosine function. Now, if we have the sum or subtraction of two cosine functions, then we end up with the product of either two sine functions or two cosine functions. If we have the sum of two cosine functions, then we end up with the product of two cosine functions. If we have the subtraction of two cosine functions, then we end up with the negative of the product of two sine functions.

So, if you have the sum of two functions, then your first term will always be the same as the functions you are adding or subtracting. Think of the plus sign as telling you that, 'Yes, I am positive that these functions are first.' Take a moment and see if you can spot other patterns.

Uses & Applications

Remembering these identities will help you when you need to solve trigonometry problems. You will use these identities to help you simplify more complicated trig problems and in proving other trig statements. In higher math, such as calculus, these identities help in solving complicated integral problems. Are you ready to take a look at a couple of examples? Okay, let's go.

Example 1

Our first example wants us to rewrite sin (35) - sin (15) using multiplication. How can we do that? We can do this by making use of our identities. We look at our list and we see that our problem matches the second identity. We can use this identity to help us rewrite our subtraction of two sine functions in terms of multiplication. We follow our identity, adding our two angles and then subtracting our two angles before dividing by 2. We get 2 cos (25) sin (10) for our answer.

sum identities

That one wasn't so bad. See how memorizing these identities makes these problems that much easier? You would have spotted this identity a mile away. Just one look at the problem, and you would immediately recognize it as a problem involving this identity.

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