Write each expression as a product of trigonometric functions....

Question

Write each expression as a product of trigonometric functions. See Example 8.

cos 5x + cos 8x

Accepted Answer

Welcome back. I am so glad you're here. We're asked to write the given trigonometric sum as a product. Our given trigonometric sum is the cosine of 13 X plus the cosine of 16 X. Our answer choices are answer choice A two cosine 29 X divided by 2 S3 X divided by two answer choice B two sine 29 X divided by two cosine three X divided by two answer choice C two cosine 29 X divided by two cosine 33 X divided by two and answer choice D cosine 29 X divided by two cosine three X divided by two. All right. So we're given a trigonometric sum. We need to write it as a product. So we're going to use the product to sum identities. Now, specifically this one is a cosign being added to a cosign. So we recall from previous lessons and we take a look at those product to some identities in the textbook. And that one is where we have one half multiplied by the quantity of the cosine of A plus B plus the cosine of A minus B. And that is equal to the cosine of a multiplied by the cosine of B. So essentially what we need to do here is figure out our A and our B and we can plug those in. But first we notice that our given trigonometric sum doesn't look exactly like the one from the product to some identities. The difference being that the product to some identity has this one half being multiplied in the front. So what we need to do is get rid of that one half. We'll do that by multiplying by two. But what we do to one side, we do to the other as well. So when we bring this down on the right hand side, this is equal to, it'll be a two multiplied by the cosine of A and the cosine of B. So now it's a matter of figuring out what's A and what's B. So we can identify that the first cosine function that's being added on the left hand side is 13 X for its angle and that's equal to A plus B. So we can write A plus B equals 13 X. And then we can identify that the second cosine function being added the angle there is 16 X and that's equal to A minus B. So we can write A minus B equals 16 X. And here we've got a system of equations. We can use this in order to solve for A and B. So I'm going to use addition to solve for this system of equations. I'm going to add these two equations together. And so we'll have A plus A is two A and the B minus B cancels out. So we have two A equals and then on the right 13, X plus 16 X is 29 X, we can divide both sides of the equation by two and we'll get an A value A equals 29 X divided by two. So we've got one of them. Now, we just need to solve for B, we can use substitution, we can plug in for A and we can just use the first equation. That's fine. Our 29 X divided by two. So we have 29 X divided by two and then plus B equals 13 X, we can subtract 29 X divided by two from both sides. And then on the left, it's just A B equals on the right 13 X minus 29 X divided by two. We want a common denominator. So we'll multiply our 13 X by two halves. So this will give us 26 X divided by two minus 29 X divided by two, which simplifies to be a negative three X divided by two. So now we can go back and plug in our A and B values into our equation. So we know that it's the two multiplied by cosine of A is 29 X divided by two multiplied by the cosine of B is negative three X divided by two. Now only one thing left, we recall from previous lessons that cosine is an even function, which means that the cosine of a negative value we'll put negative X here is equal to the cosine of that same value as positive. So a positive X. So a cosine of negative three X divided by two is the same as the cosine of positive three X divided by two. So we can bring this down. We'll have two cosine 29 X divided by two cosine positive three X divided by two. And there is our trigonometric sum written as a product. We look at our answer choices and this matches with answer choice. Oh Not B excuse me, it matches with answer choice. C well done. We'll catch you on the next one.

Write each expression as a product of trigonometric functions. See Example 8.
cos 5x + cos 8x

Key Concepts

Sum-to-Product Identities

Trigonometric Function Properties

Angle Manipulation and Substitution

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