 Welcome back to our lecture series Math 12-10, Calculus I for students at Southern Utah University. As usual, I'll be your professor today, Dr. Ange Misseldine. In lecture 21, we're going to talk about derivatives of trigonometric functions. We've talked about a lot of algebraic functions already using the power rule and also the product and quotient rules. We've also talked a little bit about the transcendentals, that is we can take the derivative e to the x. But we're now in a position where we can talk about derivatives of trigonometric functions. In order to calculate the derivatives of trigonometric functions like sine, cosine, tangent, et cetera, we need to first talk about limits involving trigonometric functions. For which one very important such limit we've already talked about in this lecture series, you will hopefully recall that previously when we learned about the squeeze theorem, we had learned that the limit as x approaches zero of sine of x over x is equal to one. If you don't remember that, by all means, click the on-screen link to take a look at the proof of that. What we wanna do is push from this right here, sine of x over x equals one. We wanna push this over and show that the limit as x approaches zero of cosine x minus one over x is likewise zero. It turns out that these two indeterminate forms will be very important as we try to calculate derivatives of trigonometric functions as these difference quotients will in fact show up there. In fact, I do wanna mention that these are difference quotients, right? Type of things you see in derivatives. Notice, of course, that if you plug in zero right here into cosine, you're gonna get cosine of zero, which is itself one, one minus one is zero over zero, which would be zero. Likewise, if you take sine of zero, which is zero, you get zero over zero, which is likewise one. So knowing this fact already established by the squeeze theorem, let's see why the limit of cosine x minus one over x is zero as x approaches zero here. So investigating this, first of all, consider our limit. What we're gonna do is we're gonna multiply the top and bottom of this ratio because we can't just plug in x equals zero right now. We're gonna multiply the top and bottom of this ratio by cosine of x plus one. And to motivate what we're doing here, think of the following scenario. If we have the square root of two over two minus one, and this all sits above pi over four, which again might seem like a weird fraction to be considering, but what have you? If this was our fraction, you're asked to rationalize the numerator, then your process would probably be like the following. Like, okay, I'm gonna take the numerator, which is a square root of two over two minus one, but I'm gonna add one instead. That has switched to sine. You'll of course, didn't notice that as you foil out the numerator, you're gonna end up with a one half the square root of two over two times itself. You're gonna get one half. Then you're gonna get a minus root two over two, a plus root two over two, and then you're gonna end up with a minus one for which those guys cancel. You're left with this one half minus one, which is a rational number you can subtract and end up with a negative one half as well and leave the dom there what it is. Well, why did I pick these numbers right here? Well, if you come to this right here, if you took X to be pi over four, then you're gonna take cosine of pi over four, which is square root of two over two. Have we done the same thing with like X equals pi over three, excuse me, pi over six, you'd end up with cosine as the square root of three over two, and we can go from there. I guess what I'm saying is because cosine and sine satisfy a Pythagorean relationship, recall that cosine squared X plus sine squared X equals one. It's very much to be understood that cosine very much behaves like a square root function. After all, cosine of X is equal to plus or minus the square root of one minus sine squared. Like I said, cosine very much acts like a square root. It does in many situations. And therefore in this situation when you have this cosine of X minus one or a cosine of X plus one, we often wanna think, well, I wanna rationalize the numerator. That helped me with many algebraic settings. Maybe it'll help me with trigonometric settings as well. And so you'll see what happens when you foil this out. Cosine times cosine is the cosine squared. Negative one times one of course is a negative one, but in all the other cases, you're gonna get cosine times one and negative one times cosine, they cancel each other out. And so the numerator becomes cosine squared minus one. The denominator, well, we don't multiply out denominators. We leave them alone. So we're gonna leave it factored as X times cosine of X plus one. So what do we do with this cosine squared minus one? Well, the whole point of rationalizing, it was that we can use this Pythagorean relationship. Notice that if we rearrange this thing a little bit, we're gonna get that cosine squared X minus one equals negative sine squared of X, which gives us the statement we see down here. In which case, then we're gonna factor this thing in the following way. The numerator of course can break up as negative sine X times sine X. And then we need another factor. So we're gonna go to times one. And then the denominator we're gonna factor as one times X times cosine X plus one. Like so, and so we're gonna put those things together. So this one goes together, this one goes together and this one goes together, which you see far less messy written right here. We're gonna take the limit of negative side of X times sine X over X times one over cosine X plus one. And we're gonna treat each of these factors differently in terms of the limit. That is the limit of a product is a product of the limit. So we're gonna take the limit as X approaches zero of negative sine, the limit as X approaches zero of sine X over X and the limit as X approaches zero of one over cosine X plus one. Now the first and last one don't have any issues with it. By continuity, we can just plug in X equals zero. In which case, you're gonna get sine of zero with sine of zero is equal to zero. And then likewise with the last one, again, by continuity, if you plug in X equals zero, you're gonna get cosine of zero, which itself is one, like we mentioned before, in which case you get one over one plus one, one plus one is two, so it's one half. So you get zero times one half. Well, zero times anything should be zero, right? Well, zero times any number. How do we know that the third limit we haven't considered yet is even defined? Ah, this is the one we did earlier with the squeeze theorem. The limit as X approaches zero of sine X over X is always equal to one, and one times zero is gonna equal zero, thus finishing the proof that we were looking for. So coming back up here, in summary here, these are two very important limits that you're gonna wanna memorize for future calculations. The limit as X approaches zero of sine X over X equals one, that's the most important one, remember that. But we also have that the limit as X approaches zero of cosine X minus one over X is equal to zero. Remember those ones, you want to memorize them in order to be successful in future limit calculations involving trigonometric functions. Let's look at two such examples. What if we wanna calculate the limit as X approaches zero of X times cotangent of X? Well, in its current form, it might not be obvious what to do, but what we can do is use trigonometric identities here. That is to say, what is cotangent? Cotangent is one over tangent. That is to say it's cosine over sine, like so. And so even if we didn't use a calculator, if we try to plug in X equals zero right now, we'd get a zero in the numerator, we'd get zero times one, but then this is zero on the bottom. This is indeterminate form, so we have to somehow remove the indeterminate form. This will come from factoring. We're gonna take the limit as X approaches zero of X over sine of X. So we're gonna put that one together and then let cosine deal with itself, well, all by itself. Because as X approaches zero, cosine's gonna go to one. There's no problem with cosine. Cosine there is just gonna become a one. The issue comes from as X goes to zero, you have an X on top which gives you a zero and a sine on the bottom which gives you a zero. So this right here is where we get zero over zero. But like we saw in the previous slide, this indeterminate form has already been dealt with. Now, admittedly it's upside down, right? But that just means that when we take the limit as X approaches zero, by continuity reasons, as X approaches zero, we're gonna end up with the reciprocal of negative of one. The reciprocal has one to the negative one power. That reciprocal, so we get one over one. You can times that by one. But of course, one divided by itself is still one. This is gonna be one at the end. So that's another thing to remember. If you ever have to take the limit as X approaches zero of X over sine of X, that likewise is equal to one because it's the reciprocal of one. So, okay, there you go. You can see how that limit comes into play here. It's very handy. Let's consider one last example in this video here. Let's consider the limit as X approaches zero of sine of seven X over four X. And notice, we've now changed the period of sine. Now, in this situation, it's like the denominator doesn't quite have what I want. This common factor of four could come out of the limit calculation. We end up with the limit, or we get one fourth times the limit as X approaches zero of sine of seven X over X. Now, this isn't quite what we had before. So we know that limit as X approaches zero of sine X over X, that'll be one. But how does the seven affect things? How does it change a period? Well, I want you to think of the following idea here. What if we make the substitution that theta equals seven X, like so? All right, so if theta equals seven X, note that as X approaches zero, that means that theta will approach seven times zero, which is equal to zero. So as X approaches zero, theta will also approach zero, although the rate of which they are approaching zero is slightly different because of that factor of seven right here. So what if we do the following? What if we multiply top and bottom by seven over seven? That's just the number one. It doesn't really change anything, right? But because of limit properties, this seven in the bottom is allowed to come inside of the limit. It's just a constant multiple. And so we end up with seven over four times the limit as X approaches zero of sine of seven X over seven X. Now we're gonna make this substitution, right? Theta is equal to seven X. So this becomes seven fourths times the limit of theta as it approaches zero. Because again, as X goes to zero, theta goes to zero. And you end up with sine of theta over theta. Oh, that's equal to one. So we get seven fourths times one, which is equal to seven fourths. Now in retrospect, it's very easy to detect what happened here. Notice we started off with a seven on top and a four on bottom, seven fourths. Basically, as we've seen before with various limits, as X is approaching zero, it turns out that sine and X approach zero at the same speed, which is why the limit turns out to be one. Now because of these stretches that have happened, these factors here, we've changed the factor. And so now sine is approaching X, excuse me, sine is approaching zero, basically with a seven to four ratio, because we sped up the sine and set up the four, or sped up the X as well. And so my point of this is that these limits, the sine X over X, cosine X minus one over X, knowing these limits helps us to calculate various trigonometric limits, which are gonna be of the form zero over zero, these indeterminate forms. These techniques memorizing these things will be crucial for us to calculate trigonometric limits. It'll also be the crux for which we can calculate trigonometric derivatives, which we'll see in the next video of our series.