Express each function as a trigonometric function of x. See Ex...

Question

Express each function as a trigonometric function of x. See Example 5.

cos 3x

Accepted Answer

Welcome back. I am so glad you're here. We're asked to write the given trigonometric function in terms of X. Our given trigonometric function is the cosine of six X. Our answer choices are answer choice. A 16 cosine to the sixth X minus 24 cosine to the fourth X plus nine cosine squared X minus one answer. Choice B 32 cosine to the sixth X minus 48 cosine to the fourth X plus 18 cosine squared X plus one answer choice C 32 cosine to the sixth X minus 48 cosine to the fourth X plus 18 sine squared X minus one and answer choice D 32 cosine to the sixth X minus 48 cosine to the fourth X plus 18 cosine squared X minus one. All right. So essentially what we're being asked to do is to get this six out of here. We want it in terms of the function of X, not of six X. And we are going to go through Kuwait the journey to get that little old six out of there. So the first thing we're going to do is we know that the cosine of six X is equal to the cosine of two multiplied by three X. And that's important because that looks a lot like one of our double angle identities. Our double angle identity here is that the cosine of two A A is a capital A for our angle here is equal to two cosine squared A minus one. So what's our A, our A is three X for this one, we're going to have different A's for different identities, but for this one, A is three X. So let's rewrite this with three X. So we'll have the cosine of two, multiplied by three X is our A equals two cosine squared three X minus one. So we've gotten that six out of there and now we still have this three. So we're actually going to spend quite a bit of time getting rid of that three. And instead of looking at the cosine squared of three X, we're going to kind of come off to the side and look at the cosine of three X, we'll get that three out of there and then we'll use substitution and plug that back in. So the cosine of three X, well, we know that the cosine of three X is equal to the cosine of X plus two X. And why is that helpful? Well, we recall from previous lessons that, that looks a lot like a cosine sum identity. So our cosine sum identity is that the cosine of a plus B is equal to the cosine of A multiplied by the cosine of B minus the sine of A multiplied by the sine of B. So what we're going to do is figure out our A and our B A is going to be X for this one and B is going to be two X. Now let's plug those in. So we have the cosine of X plus two X equals the cosine of A is X multiplied by the cosine of B is two X minus. The sign of A is X multiplied by the sign of B is two X. Now look at that, we went from A six X. Now we're down to the highest one in front of the X is A two X. So now let's work on getting rid of those two X's. We'll start with the cosine of two X for the cosine of two X. That's a lot like our double angle identity that we talked about before that the cosine of two A equals two cosine squared A minus one. What is our A here A is X. So let's substitute that in. We've got the cosine of two X equals two multiplied by cosine squared A is X minus one. Now, we can use substitution and plug two cosine squared X minus one in and we've gotten rid of one of our twos in front of the X. So let's bring this down. We've got cosine X multiplied by substituting in for cosine of two X two cosine squared X minus one, then bringing down the rest minus sine X sine two X. OK. Let's now get rid of this two in front of the sign of two X. So that one is also going to be a double angle identity. This one we haven't talked about yet. But recalling from previous lessons in the textbook, this is the sign of two A equals two S sign a multiplied by cosine A. So here, what's our A A is X again? So we can substitute that X in. This is the sine of two, not A but X equals two sine X cosine X. Let's substitute that in. So bringing things down, we have the cosine of X multiplied by the quantity of two cosine squared X minus one minus sine X multiplied by now substituting in here two sine X cosine X. OK. We got rid of all the numbers in front of our X's. Now it's just condensing everything we're going to distribute. We're going to get rid of our signs and then we'll go back and substitute in to that three X that we were working on before. So let's first distribute, we have cosine X multiplied by the quantity of two cosine squared X minus one. That's going to give us two cosine cubed X minus cosine X. And then over here, we can combine this sine X multiplied by the two sine X So we'll bring down minus that's going to be two sine squared X cosine X. Now, we've only got one sign in there and we can convert that to a cosign, so we can combine more terms. So looking for a moment at just this sine squared X, now we're going to use a fundamental identity. We recall from previous lessons that the sine squared of an angle A is equal to one minus the cosine squared of that angle. And so what's our A A here is X? And so we'll substitute that in, we have sine squared of X equals one minus the cosine squared of X. This will get substituted back into our equation. So bringing things down, we have two cosine cubed X minus cosine X minus two. That's multiplied by the quantity of one minus cosine squared X. And that's multiplied by cosine X. So now we'll do some distribution bringing down two cosine cubed X minus cosine X and then we have minus two and then that's plus two cosine squared X, but that's all getting multiplied by our cosine of X. So let's distribute that cosine X as well. So we've got, let me distribute the cosine X to the negative two and to the positive two cosine squared X. I'll do that on the next line. So we've got two cosine cubed X minus cosine X minus that two is getting multiplied by the cosine X two cosine X and then plus that'll be two cosine cubed X. OK. Let's move this to a different part of the screen and we will just combine like terms now. So we have two cosine cubed X plus another two cosine cubed X that is four cosine cubed X and we have negative cosine X minus two cosine X. So that is minus three cosine X. And if you recall what we started with was cosine of three X. So we're saying that the cosine of three X is equal to four cosine cubed X minus three cosine X. Now we're going to go back up and substitute in our four cosine cubed X minus three cosine X for our three X. So let's bring that down. We've got two cosine squared and then, well, here we'll write it this way. So it's two multiplied by our cosine of three X is going to be written as four cosine cubed X minus three cosine X. And that whole thing has to be squared. That's where the squared comes back in and then a minus one. OK. The rest of this is just some fun fun algebra. So we have to square this, we have to foil it out. So we'll have a two multiplied by the quantity of four cosine CX minus three cosine X that's multiplied by itself the quantity of four cosine cubed X minus three cosine X and don't forget minus one at the end. So I am going to foil this out very quickly. If you need to pause it to foil it more carefully, definitely go for that. So bringing down the two multiplied by the quantity of when we foil this out, it's going to be 16 cosine, six X minus, combining the outsides and the insides will give us minus 24 cosine to the fourth X and then our lasts are plus nine cosine squared X, then still minus one and all we have left to do is distribute that two in front to what's in our quantity here. So distributing that two, we'll have 32 cosine to the sixth X minus 48 cosine to the fourth X plus 18 cosine squared X minus one. And that's it. All that. There is the cosign of six X. We look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

Express each function as a trigonometric function of x. See Example 5.

cos 3x

Key Concepts

Multiple-Angle Formulas

Trigonometric Identities

Function Composition and Angle Multiplication

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