 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 Mistledine. In this video, we want to actually introduce what trigonometry is properly. That is, when one thinks of trigonometry, sure, you think of triangles and circles and such, but usually you think of sine, cosine, tangent, those trigonometric ratios. We're now in a situation which we're going to define exactly what those are. In this lecture, we will define the trigonometric ratios in two different ways and show that they're actually the same thing when the two situations overlap. In this video, we'll give you one of the definitions, and this is going to be the definition of a trigonometric ratio with respect to an angle. So imagine we have some angle theta, it's just some angle in the plane, and so we're going to put this angle in its standard position. Remember, standard position means that the initial side of the angle coincides with the positive x-axis, and then the terminal side of the angle is whatever it is. It's in the plane somewhere. So what we're going to do is we're going to pick any point on the terminal ray of the angle other than the origin itself. So the origin is, uh-uh, can't use that one. So you're going to pick some point on the terminal side, and arbitrarily we'll call this point x comma y. The x coordinates x, the y coordinates y, sure. So when you pick, and it doesn't matter which point you choose, the trigonometric functions will be based upon ratios, and so choosing different points on the ray, so long as it's not 00, will give you the same ratio that is when you put it into simplest terms. So they'll all be proportional to each other. So you pick some point on this terminal side right here. Well, when you look at this point, this, the terminal side, go into the origin, the initial side, and if you drop the line down as a perpendicular, it always forms a right triangle with one of the vertices being the origin itself. And because it's a right triangle, the Pythagorean equation applies to this triangle, which notice that the one side of the triangle is going to be this horizontal length, which is just the x-coordinate. It's how far to the right are you from the y-axis, that's what the x-coordinate measures. And another dimension is going to be the y-coordinate, this vertical line here has its, has its length, the y-coordinate, how far above the x-axis are you. And so by the Pythagorean equation, the hypotenuse of this right triangle, if we call it r, always will satisfy the relationship x-square plus y-square equals r-squared. Now, if you're wondering why r, well, the idea is r here is the radius of a circle. And that's a topic we'll talk about in the future. That turns out there'll be a third way of defining these trigonometric ratios with respect to circles. But that'll be a topic for another day. Okay, so given any point on this terminal side, we have these three numbers, x, y, and r, where x and y are the coordinates of the point itself. And then r you can calculate from the Pythagorean relationship. In particular, r is always going to be the positive number, which is the square root of x-squared plus y-squared, like so. And so what we're going to then going to do is we're going to define six ratios between these three numbers. So we define the sine ratio, denoted Sin for short, sine of theta is defined to be y over r. So we take this side that's opposite of the angle theta, the vertical distance there, and you divide it by this distance from the origin. So let's say you want to think of these numbers here, right? x measures the horizontal distance along the x-axis, y measures the vertical distance along the y-axis. You can measure that same thing right here. And then r is this oblique distance away from the origin. All right, so sine is going to be y divided by r. The second trig function, which we call cosine, it's denoted c-o-s for short, cosine of theta, this will be denoted as x divided by r. It's always going to equal that ratio. And then the third trigonometric ratio is called tangent, denoted tan of theta for short, tangent of theta. Tangent is the ratio y divided by x. So we have three other ones. The fourth one we're going to call cosecant of theta, denoted c-sc for short. Cosecant is r divided by y. Secant, denoted sec of theta, is defined to be r over x. And then the last one called cotangent, c-o-t of theta, this is defined to be x over y. And so given these three numbers, x, y, and r, when you have a fraction, the order matters, like that is the numerator and denominator. If you switch those roles, that gives you a different fraction. And so if we take these three numbers and we take this, there's only six possible fractions you can build using x, y, and r. And these trigonometric ratios get exactly each and every one of these. Now, there are a few degenerate cases I should mention. That is, there are situations when, for example, r is itself, a is the number zero, right? Like if you take a 90 degree angle, for example, then the point you take over here would be like zero comma y. So it could be that x equals zero, which then means whenever x is in the denominator, like in secant and then tangent, that actually means they would, they do not exist. It would be, they'd be undefined. We might denote this as d and e for short. Similarly, if you have a zero degree angle, turns out your angle would actually, the initial and terminal side would both be the positive x-axis here. In that situation, you pick something on the x-axis, you're going to x comma zero. Likewise, if y is zero, you'd see that anything with a y in the denominator, like cosecant and cotangent, those would actually be undefined. It would not exist in that situation. And that's actually the naming strategy going on here. That you see there's these six trigonometric functions. You have sine, secant, and tangent. And then you have a cosine, a cosecant, and a cotangent. Three of them just have the prefix co in front of the others. And this co is actually short for complementary. It has to do with complementary angles, which we'll talk about in another video. The reasons that they're labeled the way they are is you'll notice that both cosecant and cotangent have a y in the denominator. They're undefined at the same time. And then secant and tangent both have an x in the denominator. They're undefined at the same time. All right. And in this discussion, the number r is always going to be a positive number, because we're considering the positive distance away from the origin. So let me give you some examples of how these trigonometric ratios are then computed. Suppose we have the angle theta illustrated as so it's in standard position so that the point negative two comma three is a terminating point for the angle. So what this tells me already is that the value x, we're going to set to be negative two, y, we're going to set equal to three. And then by the Pythagorean relationship, r is going to equal the square root of negative two squared plus three squared, which gives us the square root of four plus nine, which equals the square root of 13. And so then if we go through the definitions, we're going to do sine of theta, sine, remember as y over r. So we're going to get three over the square root of 13. Now, you might have been taught in the previous algebra class that whenever the denominator is rational, you must rationalize it. If you did that, you would times both top and bottom by the square root of 13. And then that would then give us the ratio three times the square root of 13 over 13, which I would argue that this might not necessarily be simpler than what we had before, right? Because there's taking more symbols to describe the same ratio. The reason we rationalize fractions is not because of some simplification process. It's actually so that we can add fractions together. Since I'm not trying to add any fractions right now, that process is actually not necessary here. So I guess what I'm saying is if you don't rationalize the denominator, you will be forgiven. That's really not a transgression whatsoever. If we look at cosine theta, cosine is going to be x over r this time. So we end up with negative two over the square root of 13. You can just leave it as it is. Tangent of theta, remember, was y over x. This one turned out to be three over negative two. Or if you prefer negative three halves, it doesn't really matter where the negative sign goes as long as we have a negative sign in there somewhere. The next on our list was cosecant theta, which was defined to be r over y, which is going to give us the square root of 13 over 3. Next, we had secant of theta, which remember that was r over x, which that turns out to be the square root of 13 over negative two. Or like I said, you can write this as negative square root of 13 over two if you prefer. And then the last one, we had cotangent theta, which is equal to x over y. And so that looks like in this situation, three over negative two are, we'll say, negative three halves. That will be our six trigonometric ratios for this point. That is for this angle. Notice, we don't even know the angle measure. The trigonometric ratios are computed from the terminal point. That is a point on the terminal side. It's not actually from the angle measure itself. Now let's suppose we did know the angle measure. Let's find sine and cosine of the angle 45 degrees. Well, if you put it in standard position so that, again, the initial side coincides with the positive x-axis, then we'd have to find some point on this line segment on the terminating side. Now, the good news for 45 degrees, this is sort of like a very special case. The terminal side actually coincides with the line y equals x. This is the line for which the x and y coordinates are the same. So we can just use the point one comma one right there. So this gives us that x equals one, y equals one, and then r is going to equal the square root of two. Notice you're going to take the square root of one square plus one squared, so the square root of two. And so then we see that sine of 45 degrees, this, which sine is always going to be y over r, this is equal to one over the square root of two, or if you rationalize this, you get square root of two over two. Cosine of 45 degrees, this is going to look like x over r, which then also is equal to one over the square root of two. It's the same number. So you get root two over two if you rationalize it. And we can do the same thing for the other trigonometric ratios, tangent, which would be y over x, cotangent, which is x over y, secant, which is r over x, cosecant, which is r over y. We can compute the other trigonometric ratios. So long as you have x, y, and r and you can check the definitions, then you can handle all of these. I want to do one more example in this video, handling another one of these degenerate cases. That is, when the terminal side doesn't actually exist properly in one of the four quadrants, what if the terminal side is one of the arrows on the compass rows? That is zero degrees, 90 degrees, 180 degrees, or in this example, we take 270 degrees. How do you handle those sort of special degenerate cases here? Well, if you took 270 degrees like we are right here, the terminal side would be the negative y-axis, and so we need to find a point on that. The simplest point to choose would probably be the point zero, negative one. So we see that x is going to equal zero, y will equal negative one, and then r is going to equal the square root of zero squared plus negative one squared. This equals the square root of one squared, square root of one, we could say, and this is going to be a one. Even though x and y could be positive or negative, depending on which quadrant they're living in, the radius r will always be a positive number. And so our numbers we're doing here are zero, negative one, and one. So sine of 270 will be y over r, which simplifies to be negative one. If you do cosine of 270 degrees, you're looking at x over r, which looks like zero over one, which is zero. If you do tangent of 270 degrees, this will be y over x, which you end up with negative one over zero. Like I mentioned before, if you get a zero in the dominar, it turns out tangent is undefined at 270 degrees. We sometimes say d and e if it does not exist. Similarly, if you were to do secant of 270 degrees, secant, you're supposed to be looking at r over x, which again you get a zero in the dominar. So secant does not exist for 270 degrees. Likewise, tangent and secant are undefined at 90 degrees. Co-secant, this is going to be r over y, for which in this case you get one over negative one. That also becomes negative one. Notice that secant and sine actually equal to each other at 270. They'll also agree with each other at 90 degrees. And then lastly, cotangent, this is going to look like x over y, which is equal to zero. And so in this bizarre case, it turns out cosine and cotangent are also equal to each other at 270 degrees. This will also be the case for 90 degrees. The other case to consider besides 270 and 90, also consider the special cases of zero degrees and 180 degrees, but I'll let the viewer consider those degenerate cases there.