 Welcome back to our lecture series math 1060 trigonometry for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Missildine. In our previous video, we learned about one-to-one functions, invertible functions, that is the inverse of a function. So that, you know, if you have a function like y equals x plus three, then it has an inverse function y equals x minus three, so that when you compose these things together, let's call this one f of x, and then this would be f inverse of x. We have the property that when you do f compose with f inverse, you're gonna end up with x minus three plus three. When you do the two things together, it's as if nothing happened at all, they cancel each other out. So inverse functions are very important because they help us undo mathematical functions. But this is only possible when you have a one-to-one function. The idea is when you reflect it across the diagonal, you get a function which is also one-to-one. And you get issues like with a parabola, right? If you take y equals x squared, this function's not a one-to-one function because it fails this vertical line test. And so there's not a proper inverse function to the square root, but there is something that kind of acts like it, like sort of like a pseudo inverse. That is if we basically just erase the left-hand side of the graph, that is we limit the domain just to be when x is greater than or equal to zero, then when we reflect this graph, we end up with the square root graph, y equals square root of x. And so in this situation, the functions or inverses of each other, x squared is an inverse to the square root of x. But there's a caveat. Because the original parabola was not one-to-one, it was actually two-to-one, whenever you solve an equation using the square root, turns out there actually could be multiple answers. Like when you look at something like x squared equals four, we know, okay, to solve for x, I'm gonna take the square root of both sides, but this is the caveat here. We don't take the square root of both sides per se. We have to take the plus or minus square root. So there's two solutions, plus or minus two, which oftentimes we remember this positive square root of four is two, no problem. But we often forget the negative. And that's because we're only working with the right side of the parabola. The reason I bring this up is because when it comes to trigonometric functions, the same problem is present. Coming up with an inverse function for trigonometric functions is not as easy as maybe a one-to-one function because not a single trigonometric function is one-to-one. You can see on the screen right now a typical sine wave. So you see it right here, the sine functions as they're periodic, they go up and down, up and down, up and down forever and ever and ever, right? And so if you analyze any section of this graph, it's not gonna pass the horizontal line test, it's not a one-to-one function. But kind of like the square root versus the parabola, if we were to restrict the domain to some one-to-one region, then we could find an inverse for that section of the graph. Like what if we just take this right here, which this is often referred to as the principal region of y equals sine of x? The reason we call it that is if you look at this, if you look at just this sector right here, that are colored in green, yeah, I've restricted the domain, I've restricted my attention just to be negative pi halves to pi halves or negative 90 degrees to 90 degrees if you prefer the degree measurement there. And why did I do that? Well, there's really two reasons why we were gonna restrict the domain to be this sector right here. So the first part is that this is connected, right? This is one continuous strip. There's no discontinuities, no breaks, no holes, no gaps in the graph whatsoever. We could have tried to do something like, I'm gonna take this piece over here and this piece over here, that would create a one-to-one function, but it wouldn't be connected. We want it to be continuous, that's the first goal. And so kind of, and also central to the origin, and it does include zero, that's kind of a nice goal. But the other part is if you notice this piece of the graph contains all of the y-coordinates, this point over here when x equals pi halves is y equals one. This point over here when x equals negative pi halves, that's y equals negative one. And we get every point on the y-axis between one and negative one. So we've restricted the domain, but we've retained the entire range. So only the domain got smaller, the domain got restricted to negative pi halves to pi halves but the range stayed the same. And so on this function right here, this principal branch of y equals sine x, we then construct the inverse function for that, for which if we were to reflect this across the diagonal, the diagonal y equals x right here, so you reflect it to the side, you're gonna get something that looks like this. And this function we refer to as y equals arc sine of x, or sometimes it's denoted sine inverse of x. You put this negative one superscript next to the sine. Now you have to be cautious here, because when you see something like x to negative one, you might be tempted to think this is a reciprocal because hey, negative exponents could mean that. This is in mathematics, well, not just in mathematics, but in languages as well we refer to as a homograph. That is we're using the exact same notation to write two different meanings. Like if I write the English word, W-I-N-D here on the screen, what is the word here? Well, I promise you whatever the word you said out loud right now, I would have picked the other one. Is it wind, is it wind? Out of context, you can never know which word it is, which is why I tricked you, ha ha ha ha. But the mathematics has these same linguistic limitations. In this context, when we write sine to the negative one, we never mean reciprocals, no, no, no, no. We're never gonna mean this in trigonometry, although we mean that in algebra all the time. When you write a sine to the negative one, we're talking about like inverse functions, F to the negative one, doesn't mean the reciprocal of F, that means the inverse of F. And so sine inverse, sine to the negative one will always mean arc sine, the sine inverse function. If ever I wanna take the reciprocal of sine, if I wanna take one over sine, there's already a name for this in trigonometry, it's called cosecant. If I ever wanna talk about the reciprocal of sine, I'll say cosecant. If I wanna talk of the inverse of sine, I'll use sine inverse like you see right here. So sine inverse is a function, oh, I can leave that on the screen, sine inverse is a function whose domain is gonna be negative one to one and its range is going to be negative pi halves to pi halves. And this is the switching things around, right? The domain of sine becomes the range of sine inverse and the range of sine, which is gonna be negative one to one, this becomes the domain of the inverse function. Inverse functions flip these things around. So when you see an expression like sine inverse X of Y, what you wanna interpret that is sine inverse of X equals Y, exactly when sine of Y equals X. Or how I actually like to think of it is you like move the sine inverse to the side and you end up with X equals sine of Y. It's how you solve for inverse relationships like so. And so the important thing you wanna take away from this is that sine inverse is the inverse operation of sine, but we have to restrict the domain because sine is not normally a one-to-one function. If we restrict to the principal branch of negative pi halves to pi halves, so basically we're taking quadrant four and quadrant one, if you think of in terms of unit circles, that then becomes the range of sine inverse and we can only take in numbers negative one to one. So when you take something like sine inverse of X, you wanna think of this as the ratio, excuse me, you wanna think of X here as the ratio, that's what I meant to say. X is the ratio because after all sine of Y equals X and Y, Y here is the angle or more specifically sine inverse of X. This is always equal to an angle. So you can only accept ratios that go between negative one and one because if your ratio is too big or too small then that's not an acceptable sine ratio. Sine's always between negative one and one so sine inverse will only accept numbers between negative one and one and then it will output an angle in the first or fourth quadrant from negative pi halves to pi halves. We can play the same game for the other trigonometric functions. So take cosine for example, Y equals cosine, it's graph looks just like sine, actually it's just equal to a horizontal shift of the graph, it's also not one to one but let's grab the principal branch, that is we're looking for a piece of the graph that's connected continuous near the origin, near X equals zero and it takes all of the Y values in the range of cosine which cosine also will range from cosine of X, it'll range from one to negative one just like sine does. And so the principal branch we're gonna grab is gonna be this piece right here. We're gonna go from zero to pi that is we're gonna grab quadrant one and quadrant two when it comes to cosine. And so then when you invert this to reflect this across the diagonal line you get a picture that looks like this. This is what we mean by arc cosine or sometimes called cosine inverse for short. All right, the domain of arc cosine will just be like sine, it'll range from negative one to one. The range on the other hand is gonna equal zero to pi, those aren't the same regions. Sine inverse will always output something between the fourth and the fourth and first quadrant. Cosine inverse will always give you something the first and second quadrant. It has to do with the nature of the original graph. We could play a similar game for inverse tangent or sometimes called arc tangent. Now when it comes to arc tangent, arc tangent, the range is actually all real numbers. It goes from negative infinity to infinity because of these vertical asymptotes that are on the graph of tangent. We want a connected piece that grabs all of the points and we also want to be kind of near X equals zero. So the principal branch is actually gonna be this connected piece right here. It's gonna go from negative pi halves to pi halves. So just like sine, we're gonna end up with the fourth quadrant and the first quadrant, like so. But the domain is a little bit different. This is a typo there, whoopsie daisy. This should be less than, less than right there. That is to say when it comes to the domain of, when it comes to the range of tangent, excuse me, we're gonna restrict the domain of tangent here to be negative pi halves to pi halves. That is pi halves and negative pi halves are not included because tangent is not defined at those values. The range is gonna be all real numbers, negative infinity to infinity. Then when we flip this thing around to get the inverse function, we end up with tangent inverse or arc tangent as it's sometimes called, same thing here. It's domain is gonna be all real numbers because the range of tangent was all real numbers, but it's range, the range of arc tangent will be restricted to negative pi halves over pi halves here. That is arc tangent will have two horizontal asymptotes as x approaches infinity, positive infinity. We're gonna see that x, or excuse me, y approaches pi halves. And it does it from below, but we don't need to worry about that. It'll approach pi halves. And then as you look at the other horizontal asymptote, it has two horizontal asymptotes. As you approach negative infinity here, as x approaches negative infinity, we see y would approach negative pi halves from above precisely, but we don't need that level of precision in the situation. Now, unlike when we graph sine and cosine and tangent previously, we're not gonna worry about graph transformations for sine, arc sine, arc cosine, arc tangent. Although you could do those various things. It's important to have a basic idea of what these graphs look like, but we're not gonna put as much focus on the graphs of these things. Mostly just wanna understand that they're inverses of the trig functions we've began with. Perhaps the only exception of that, it's gonna be arc tangent. We won't be graphing arc tangent a lot, but this function that's always increasing with an upper bound and a lower bound have these two horizontal asymptotes. This is not a property you see a lot with like rational functions or polynomials, maybe in an algebra class. So for mathematical modeling purposes, arc tangent's a very nice function to play around with, but that's not something we're gonna really worry about at this venture. And I should mention that similar statements can be said for arc secant, arc cosecant, and arc tangent. Although we won't do all of the details of those things, by analog, we can mimic what we've done in this video and define precisely what one means by arc secant, arc cosecant, and arc cotangent. We'll stick mostly with arc sine, arc cosine, and arc tangent, mostly because amongst other reasons, we don't need all of the inverse functions we can get with the three we have. And also your calculator probably only comes equipped with arc sine, arc cosine, and arc tangent.