Write each expression as a sum or difference of trigonometric ...

Question

Write each expression as a sum or difference of trigonometric functions. See Example 7.

5 cos 3x cos 2x

Accepted Answer

Welcome back. I am so glad you're here. We're asked to use a product to sum formula to write the given trigonometric expression as a sum or difference. Our given trigonometric expression is nine cosine five X cosine eight X. Our answer choices are answer choice A nine halves multiplied by the quantity of the cosine of 13 X minus the cosine of three X. Answer choice B nine halves multiplied by the quantity of the cosine of 13 X plus the cosine of three X. Answer choice C nine halves multiplied by the quantity of the sine of 13 X plus the cosine of three X and answer choice D nine halves multiplied by the quantity of the cosine of 13 X plus the sine of three X, all right. So the product to some identity is what we want to use here because we can see that two cosine functions are being multiplied by each other, but they have different angles. So the product to some identity that we're going to use is that the cosine of angle A multiplied by the cosine of angle B is equal to one half multiplied by the quantity of the cosine of A plus B plus the cosine of A minus B. And so what we're given is very similar to that, except the whole thing is being multiplied by this nine. So we want to go to our trigonometric identity and multiply both sides by nine. And on the right, when we multiply one half by nine, that's going to be nine halves. So bringing this down, we'll have nine cosine five X cosine eight X equals nine halves multiplied by the quantity of the cosine of. Now, let's identify our A and B. Our A is that first cosine angle that's five X and our B is the second cosine angle. That's eight X. Now we can plug those in. So the cosine of A plus B A is five X plus B is eight X plus the cosine of A minus B. So that's five X minus eight X. And now it's just simplifying. So on the right hand side, we'll bring down that nine halves multiplied by the quantity of the cosign of five X plus eight X is 13 X plus the cosine of five X minus eight X is a negative three X. Now we can take a look for a moment at this cosine of negative three X. We recall from previous lessons that cosine is an even function, which means that the cosine of a negative angle, we'll say a negative X is equal to the cosine of that same angle, but positive a cosine of positive X. So the cosine of negative three X is going to be the same as the cosine of positive three X. So we can bring down the rest, we have nine halves multiplied by the quantity of the cosine of 13 X plus the cosine of three X. And there's our trigonometric expression written as a sum. We look at our answer choices and this matches with answer choice. B well done. We'll catch you on the next one.

Write each expression as a sum or difference of trigonometric functions. See Example 7.
5 cos 3x cos 2x

Key Concepts

Product-to-Sum Formulas

Trigonometric Function Properties

Algebraic Manipulation of Trigonometric Expressions

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