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 4x

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 eight X. Our answer choices are answer choice. A 128 signed to the eighth X minus 256. Signed to the sixth X minus 160. Signed to the fourth X minus 32 S squared X plus one answer choice, B 128 signed to the eighth X minus 256. Signed to the sixth X plus 160. Signed to the fourth X minus 32 sign squared X plus one answer choice C 128. Signed to the eighth X plus 256. Signed to the sixth X plus 160. Signed to the fourth X minus 32 S squared X plus one and answer choice D 128 signed to the eighth X minus 256. Signed to the sixth X plus 160. Signed to the fourth X plus 32 sign squared X plus one, right. So essentially in all of that, what we're being asked to do, getting this trigonometric function in terms of X is to get rid of the eight. So we're going to use a bunch of identities and substitutions in order to get that eight out of there. And so the first thing we'll do is we know that the cosine of eight X is equal to the cosine of two multiplied by four X because two multiplied by four X is eight X. And now that looks a lot like one of our double angle identities. We recall from previous lessons, the double angle identity, that the cosine of two A equals one minus two sine squared A. And I'm using this identity because all of our answers are in terms of S so we can identify our A A here is four X. Now we can go back and fill in for A. So the cosine of two multiplied by four X is equal to one minus two sine squared, not A but four X, all right. So we got the eight out of there. Now we're down to a four. Now we're just going to do a bunch of work to get that four out of there. We're going to take a look at this sine squared four X and just look at the sine of four X at first. Then we're going to find what that's equivalent true to go back and plug that in and square it because it's sine squared. So sine of four X is equal to the sine of two multiplied by two X. And that's very similar to the double angle identity where the S of two A equals two sine A cosine A. And so we can identify our A again A here is two X. Now let's go fill that in. So the sign of two multiplied by two X is equal to two sine, not A but two X cosine not A but two X. Now we can go plug that in. So we can bring down the one minus two. And now instead of the sine squared to four X, that is two S two X cosign two X, but that's squared because we had a sign squared. So next, everything in there in the parentheses is being multiplied so we can square everything in there. So bring down one minus two open parentheses. Two squared is four sine of two X squared is S squared two X and the cosine of two X squared is cosine squared two X. Now, we can multiply that negative two by the four to get rid of the parentheses. So this is one minus two, multiplied by four is eight sine squared, two X cosine squared two X. OK. We're down to a two in front of our X's. So now let's do some work getting rid of those twos. Let's zoom in on that S squared of two X very, similarly, we're just going to look at the sign of two X find what that's equivalent to then go back in and square that. So we can use the same a double angle identity here, the sine of two X is equal to the sine of two multiplied by X. And here our A is X. So because the sine of two A equals two sine A cosine A, the sine of two X equals two sine, not A but X cosine X A, we got rid of that too. So let's bring down one minus eight. Now that eight is being multiplied by, instead of sine squared of two X, that is two sign X cosign X all squared. Now let's take a look at this cosine squared of two X. So the cosine of two X and then we'll go back plug in and square it, the cosine of two X is equal to the cosine of two multiplied by X. And this is like our double angle identity, the cosine of two A equals one minus two sine squared A. So our A is X now plugging in X for A, the cosine of two X equals one minus two sine squared of X. So we can put this in to the one we're working on. So one minus eight multiplied by two sine X cosine X squared is being multiplied by the quantity of one minus two sine squared X. But all of that is squared because it's substituted in for the cosine squared of two X. OK. Now we can do some simplifying. So we'll bring down one minus and then we'll have the eight multiplied by. And now everything in two sine X cosine X can be squared because they're all being multiplied. So two squared is four sine X squared is sine squared X and cosine X squared is cosine squared X. And now this other 11 minus two sine squared X all squared because there's subtraction in there, we need to foil that. So lets write this one out for foiling. We've got the quantity of one minus two sine squared X multiplied by the quantity of one minus two sine squared X. OK. So now we can simplify a little bit. Bring this down. We have one minus the eight is being multiplied by the four. So that's one minus 32 sine squared X cosine squared X. We do need to foil out the one minus two sine squared X being multiplied by itself. So we can do that put parentheses for our first one, multiplied by one is one outsides plus our insides are going to give us a minus four S to the fourth X. And then our lasts will give us a plus fours. Oops, sorry, the outsides plus the insides were one or were minus four sine squared X and then the lasts are plus four S sign to the fourth X. OK. Now there's a lot of multiplication that needs to happen here. But before we get too far in that, let's take a look at this cosine squared X. All of our answers are in terms of sine. So let's convert that into sine using the fundamental identity that the cosine squared of an angle. We'll say theta is equal to one minus the sine squared of that angle theta. So we can substitute that in bringing this all down, we have one minus 32 sine squared X. Now being multiplied by the quantity of one minus sine squared. And here our theta is X and then that's being multiplied by the quantity of one minus four sine squared X plus four sine to the fourth X. OK. Now the rest of this is just lots of multiplication and bringing things together and simplifying. So we can start off distributing this negative 32 sine squared X to the quantity of one minus sine squared X. We bring down the one and then negative 32 sine squared X multiplied by one is minus 32 sine squared X and negative 32 sine squared X multiplied by a negative sine squared. X is going to be a plus 32 S to the fourth X. Now that's all being multiplied by this quantity of one minus four sine squared X plus four sine to the fourth X. So I'm going to bring this to a different part of the screen where we've got a little bit more room and now we can start distributing, we need to distribute this. Um Let's see, we need to distribute everything except that one at the beginning, I should have parentheses around the negative 32 sine squared X plus 32 S sign to the fourth X. So we're distributing those two terms to the one minus four sine squared X plus four S sign to the fourth X. So we'll start with distributing the negative 32 sine squared X bring down this one in front and then negative 32 sine squared X multiplied by one is minus 32 sine squared X negative 32 sine squared X multiplied by a negative four sine squared X is going to be plus 128 S to the fourth X negative 32 sine squared X multiplied by a positive four sin to the fourth X is going to be a minus 128 S to the sixth X. And now we can distribute this positive 32 S sign to the fourth X to the three terms, one minus four sine squared X plus four signs to the fourth X. So negative or excuse me, positive 32 sign to the fourth X multiplied by one is plus 32 sign to the fourth X, 32 sign to the fourth X multiplied by negative four S squared X is negative 128 S to the sixth X and 32 signed to the fourth X multiplied by four, signed to the fourth X is going to be plus 128 sign to the eighth X. All right. Now, all we have to do is put these in order and combine like terms. So the highest degree here is the sign to the eighth. So we'll start off with 128 sign to the eighth X and it's just going to mark that, that one's done. And we've got negative 128 signed to the sixth X minus another 128 signed to the sixth X combined. Those are minus 256 sign to the sixth X marking those two as done next in degree is signed to the fourth. We have 128 signed to the fourth X plus 32 signed to the fourth X. That's plus 160 signed to the fourth X marking those two as done. And then we've got just minus 32 sine squared X and then plus one and that's it. That is the cosine of eight X, I mean, clearly, right. That's what you think of when you think of the cosine of eight X. So we look at our answer choices and this is going to match with answer choice. B well done. We'll catch you on the next one.

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

cos 4x

Key Concepts

Multiple-Angle Trigonometric Functions

Double-Angle and Power-Reduction Formulas

Trigonometric Identities and Algebraic Manipulation

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