 Welcome to our lecture series Math 1060, Trigonometry for Students at Southern Entire University. For this course, I will be your instructor. My name is Dr. Andrew Misildine. Nice to meet you all. Like the name suggests, this lecture series is about trigonometry, but what is trigonometry? That's what I want to mention in this introductory video. Well, trigonometry is in fact a subset of the mathematics known as geometry, which mathematics, excuse me, geometry is the mathematics of shape and space, those type of things. So as you've studied geometry in the past, you probably learned about shapes like triangles, circles, rectangles, parallelograms, and related topics to that. Trigonometry in this course is gonna focus on planar geometry. That is, we're only gonna be doing two-dimensional things. Many of the trigonometric notions we've learned about in this lecture series could be generalized to three dimensions, four dimensions, five dimensions, and beyond, right? But we are just gonna focus primarily just on two-dimensional type problems, so-called planar geometry. But also trigonometry is not focused on all aspects of geometry. As the word trigonometry itself translates, if you look at the word try, here means three, in this case, gone is talking like a polygon, which has many sides. So first of all, and then the metric here, meter, this means the measurement of, so trigonometry literally translates as the measurement of a triangle or a trigon, all right? So trigonometry is mostly gonna be focused on triangles. And in this introductory video, I'm gonna introduce some vocabulary related to triangles that'll be essential for this lecture series. But trigonometry likes to approach geometry from the perspective where we use coordinates and algebraic equations to help us solve geometric problems. So, so-called analytic geometry. That is the coordinate geometry using the equations to study geometry. So we're not gonna be mostly focused on like geometric constructions, we're not gonna do that, we're not gonna be really focused on geometric proofs, which other geometry classes might do. Trigonometry is mostly analytic geometry focused on triangles and related subjects such as angles and circles. So let me remind you of some important geometric terms that'll be bedrock for the rest of this lecture series. So we're gonna talk a lot about points. So we take points A, B inside of the plane. Like I said, we're only gonna talk about planar problems in this course. And so let's say we have two points. Let's just draw them here on the screen. We have two points A and B. The line segment, excuse me, that connects the two points will be denoted by A, B with the line already above it, excuse me. If you don't put the line above it, this actually measures the length of the line segment. This is a common misconception here. A, B with the line above it represents the line segment that connects the two points. And then without that line, like I said, is the measure of the line. The two are quite related to each other, which is why people sometimes conflate the issues together. So A, B right here is the line segment connected by the points A and B. And so if I were to draw two lines for a moment, let me, let's consider two lines right here. And let's say that these are, these of course are lines. And so we get maybe the point A right here, B right here. If we want the whole line determined by the points A and B, we might say this as like A, B with a erode line segment above it. So this would be the line determined by A and B. Let's say we take a second line, two points C and D, then this would be the line C, D that is the line determined by C and D, all right? And so we know from basic geometry that two points determine a line. So a line is uniquely determined by two points. Well, a pair of lines that'll be extremely important in the trigonometry course is the idea of perpendicular lines. So we say that the line, a line or the line segment, all right, it doesn't have to be the whole line either. If we take the line segment A, B, we say it's perpendicular to C, D. If the angle formed between the two lines is a 90 degree angle, okay? Angle measure is something we're gonna talk about a little bit later on in this course. So how do we describe right angles or perpendicular lines without 90 degrees? Just so you know, a right angle is the angle between two perpendicular lines. And this is often denoted as we draw a little square right here. Let me make it look more like a square. And the reason we do that is that a square itself has four right angles. We often draw a little square in right angle to indicate that this is a right angle. So if we wanted to find right angles without any notion of angle measurement, we can do this with slope, for example. So if we take two lines, let's say we have two perpendicular lines, like so. Again, we have this right angle. We take the slope of one line and then we'll take the slope of the other one, call it m prime. It does so happen that perpendicular lines have the property that if you multiply together their slopes, you take m times m prime that's equal to negative one. So that's gonna be our definition of perpendicular lines for the most part. If we need a proper definition, and we'll denote perpendicular lines by drawing this sort of like upside down T. And this is supposed to be resemblant of this picture we see over here. So perpendicular lines, they're products, the product of their slopes will be negative one. That is the perpendicular slopes will have negative reciprocals, negative reciprocal slope. We'd say that three points are collinear if there's a single line that contains all three points. The significance of the constellation Orion in the Northern Hemisphere is that the three stars which we call Orion's belt are collinear in the sky and relatively close to each other, which is not something you usually see with stars. Any two points, of course, are collinear. Like I said, any two points determine a line. And so if you have a set of three or more points, if that's collinear, that's kind of significant. Well, if something's non-collinear, it means that the three points cannot be drawn on a single line. So you get something like this. And if you take three non-collinear points, this is what we mean by a triangle. So if you draw the line segments that connect these three non-collinear points, let's call this A, B, and C, the points. Then if you join together those three lines, we get a so-called triangle. So that's what we mean by a triangle. It's this shape in the plane determined uniquely by three non-collinear points. We often denote a triangle as a triangle symbol ABC. And so we're gonna talk about triangles all the time. A little bit of vocabulary you should be aware of with triangles. If all three sides of the angle are the same, excuse me, if all three sides of the triangle are the same, we call it an equilateral triangle, equilateral here, meaning that the laterals, that is the side lengths are all the same. We say that a triangle is isosceles if two of the sides are equal to each other. Now I should mention that if three of the sides are equal to each other, that implies that two of the sides are equal to each other as well. And so every equilateral triangle is an isosceles triangle, but we can get a sosceles triangles which are not equilateral. For example, if we draw an isosceles triangle like this, you have like maybe a very skinny angle right here, these two sides are congruent, but maybe the third side it's not, all right? Lastly, if all the sides have different lengths, we call that a scaling triangle, all right? Now typically when people talk about sosceles triangles, they typically don't mean an equilateral because if all three sides were the same, I wouldn't read you to say that, but be aware that by definition, the family of a sosceles triangles does include equilateral triangles, but scaling would be the complement. A scaling triangle would be the complement there of an isosceles, that is scalings don't have any of the three sides together. Later on when we talk more about angle measures, these definitions will make a little bit more sense, of course, we'll explore all this in the future, in this lecture series. If all of the angles in a triangle are acute, where acute angles are gonna be those less than 90 degrees in measurement, we call the triangle acute. Likewise, if there is at least one obtuse angle in the triangle, we call it an obtuse triangle. Obtuse here, of course, measuring, meaning that the angle measure of an obtuse angle is greater than 90 degrees. If a triangle has an obtuse angle, we call it an obtuse triangle. Now, because the angle sum of a triangle will always equal 180 degrees, a triangle cannot have more than one obtuse angle. So if has one, it's gonna be unique. And similarly, we say that a triangle is a right triangle, if it has a right angle. And if a triangle has a right angle, that right angle will necessarily be unique. And so because of that, we often reference triangles based, a right triangle based upon that right angle. So a sketch of a right triangle might look something like the following, where you see that right angle going on over here. And so we often reference things in the triangle with respect to that right angle. The side that's opposite the right angle will be the largest side of the right triangle. It's called the hypotenuse. It would be this side right here. And then the other two sides, which are gonna be adjacent to the right triangle, they're gonna be shorter in length. These are called the legs of the right triangle. So we have one leg right here, and we have another leg right here. And so again, this is some vocabulary that you should know about triangles. In trigonometry, we're gonna be particularly interested in right triangles, at least at the beginning of our lecture series. Later on, we're gonna consider non-right triangles, which are commonly referred to as oblique triangles. I mean, they could be acute. They could be a two, so they're just not right. But for the immediate part of our lecture series, we're gonna focus on these right triangles. And that leads us to the Pythagorean equation, which we'll talk about in the next video.