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.

8 sin 7x sin 9x

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 a difference. Our given trigonometric expression is 12 sign of 13 X sign of 17 X. Our answer choices are answer choice A four multiplied by the quantity of cosine of four X minus cosine of 30 X. Answer choice B 24 multiplied by the quantity of the cosine of four X minus the cosine of 30 X. Answer choice C six multiplied by the quantity of the cosine of four X plus the cosine of 30 X and answer choice D six multiplied by the quantity of the cosine of four X minus the cosine of 30 X. All right. So using product to some identities, if you recall from previous lessons and take a look at the chart in the textbook here, we've got a sign of an angle being multiplied by a sign of a different angle. So we are going to use the product to some identity where we have the sign of an angle A multiplied by the sign of an angle B and that is equal to one half, multiplied by the quantity of the cosine of A minus B minus the cosine of A plus B. And so we just need to do a couple of things here. First, we see that our given trigonometric expression looks a lot like that, but it's a little different because we have it being multiplied by this 12. And so what we do is we are going to multiply the product to some identity by 12. And what we do to one side, we do to the other as well. And when we multiply one half by 12 that gives us six. So our 12 sine 13 X sine 17 X is equal to six multiplied by the quantity of the cosine of. And here, the other thing we have to do is just identify our A and B and plug those in. So our A angle, the first sign is 13 X minus the B angle. Thats the second sign angle thats 17 X and then minus the cosine of A is 13 X plus B is 17 X. All right. Now, it's just simplifying what we have. So on the right, we have six multiplied by the quantity of the cosine of 13 X minus 17 X is a negative four X minus the cosine of 13 X plus 17 X gives us a positive 30 X. Now we note here that the cosine of negative four X. Well, cosine, we recall from previous lessons is an even function. Which means that if we have the cosine of a negative number, so we'll say negative X that's equal to the cosine of that same angle but positive the cosine of a positive X. So we can rewrite this, we have six multiplied by the quantity of the cosine of instead of having a negative four X, that's the same as four X because cosines even and then we can bring down minus the cosine of 30 X. And there we have rewritten our given trigonometric function, excuse me, expression as a sum or difference. This one's a difference. We look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

Write each expression as a sum or difference of trigonometric functions. See Example 7.
8 sin 7x sin 9x

Key Concepts

Product-to-Sum Formulas

Trigonometric Function Properties

Algebraic Manipulation of Trigonometric Expressions

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