 Hi, I'm Zor. Welcome to Unisor Education. We will talk about triangles and the trigonometry of right triangles. In the previous lectures, I did two things actually. First, I explained what's the roots of trigonometry and the roots are in the triangles, primarily in the right triangles, and basically defined functions of angles, of acute angles, as ratio between an opposite catheters of the right triangle with this acute angle. Two hypotenuse, that's the sine and the cosine is ratio of adjacent catheters to the hypotenuse. And then all other functions also were defined this way. Now, these definitions were fine for acute angles, but obviously they were not good for any angle. And then I expanded the definitions using the unit circle, the one which has the radius one, and the center in the center of coordinates and just said that an abscissa of this angle is basically, it's a cosine and originate is a sine of this angle. Regardless of what the angle is, it can be acute or it can be something like this, for instance, in which case this is the point and it also has its own x and y coordinates and the x coordinate will be a cosine of this angle and y coordinate will be a sine of this angle. Now, now we're back to right triangles, in which case I can actually use both definitions, but the first one, which was actually defined for right triangles is obviously much more appropriate and the equivalence of these two definitions we did discuss in the previous lectures. All right, so I will again go back to the original definition of trigonometric functions for right triangles as ratio between the casualties and hypotenuse and casualties between themselves. Okay, so that being said, let's just consider that we have a right triangle. I will use capital legend letters for vertices. I will use the corresponding Greek letters for angles. 90 degree is pi over 2 in regions and obviously both alpha and beta are acute and obviously their sum is also 90 degree. Some of these two acute angles in the right triangle. So all these properties are known properties of right triangles so I'm not going to prove anything. If you need the proof, you can find it in geometry part of this course. Now, besides, I will use lowercase legend letters. Across the A, I will use a lowercase. Across the B, I will use B lowercase and C. Now, what do we know about this particular triangle besides this? We'll also know the Disagarran theory. So that's about it about the triangle. Now about trigonometry. Definition sets that sin of alpha is equal to opposite casualties to hypotenuse. That's the definition. And again, this definition is completely in correspondence with definition of the function sin or in other functions which I did using the unit circle. But this definition is true for acute angles only. That definition for wherever using the unit circle is good for any kind of angle. But for acute angles, they are the same. So, cos sin is B over C. So, these are two major functions. The third function which is important but not as major. Tangent is A over B, opposite towards adjacent. Now, then there are three other functions which are used less than these. Quot engine is B over A of this angle. And two more functions. Second is C over B. And cos second is C over A. Now, I have to tell right up front that sometimes engine is abbreviated like TG, not as TAN. And cos second sometimes is cos second. It depends. But different authors prefer different things and different countries actually prefer different abbreviations. These, by the way, are more used in Europe. In America, these are more used. I'll use these ones. It doesn't really matter. Okay, so these can be considered just as a definition. There is nothing to prove here. However, what immediately follows from here that these two major functions actually are sufficient to derive anything else. Tangent is obviously sine over cosine. A over C divided by B over C. C would be reduced, so it would be B over A. Now, this is obviously an inverse. Cosine over sine. Now, this is obviously one over cosine. This is one over sine. So, as you see, these functions do play the major role and, quite frankly, in all the practical implementations and practical calculations, it's usually these two functions which are really participating. They're never something else usually derived. Well, in many cases tangent also participates, but that's about it. Cos tangent second and cos second are very rarely used. They don't really matter. What does matter is that these can be derived and basically that's it. So we will concentrate on these functions and maybe sometimes on tangent as more frequently used. Now, what follows from here is... I don't need this anymore. What follows from here is... Now, you remember that these functions are functions of angle only, which means if you will take any other right triangle with the same angles alpha and beta, then the ratios will be exactly the same, which means that we can just tabulate the values of these functions, actually only these two functions, for any angle, whatever we want, and then just use these tables or these pre-calculated values which are already built into our calculators and computers. We can use them to resolve certain right triangles in terms of if you have one side, how to find another side. Well, here is how. I mean, it just directly follows from these formulas. Now, from this formula, for instance, what do we have? A is equal to C times sine alpha. So, if you have a hypotenuse and an angle, then you can get a calculus. Or you can get another calculus. It's equal to cosine. Similarly, how to find hypotenuse, for instance? Well, it's A divided by sine alpha or B divided by cosine alpha. So basically, any kind of manipulation with these formulas, you can resolve one over another if you know the... I didn't really mention how to resolve A if you know B. Well, that's from tangent, right? So, A is equal to B times tangent alpha from here. So, in any case, you can get an angle of a right triangle, an acute angle, I mean, obviously. And one side, you can, using trigonometric functions, determine the value of the lengths of another side. Either it's another calculus or it's a hypotenuse, whatever it is. All right. So, this is how we will use these trigonometric functions in practical implementation, whatever the problem is. Okay. Here is a very important quality. Now, we know that sine of alpha is an opposite calculus over hypotenuse. Now, what's a sine of beta? Well, same thing. Opposite calculus, which is beta, which is B over hypotenuse, and cosine beta is adjacent. Basically, that's all you have to remember. I mean, if there is anything to remember about trigonometry, it's just the definition that the sine is opposite towards hypotenuse and cosine is adjacent to the angle over hypotenuse. That's basically it. Everything else is following logically. But one of the things which follows quite logically is this. Compare this, B over C, and this. So, what it means is, sine A is equal to cosine beta, and cosine alpha is equal to sine beta. This is the same as this, and this is the same as this. So, if two angles complement each other to 90 degrees, like in this case, two acute angles complement each other to 90 degrees, then the sine of one is equal to cosine of another, and cosine of the first is equal to sine of another. That's very important. Therefore, for instance, sine of 40 degrees is equal to cosine of 50 degrees, because 40 and 50 is 90. So, that's very important quality which you have to remember. And the only thing which remains actually for me to discuss in this lecture is just to find what the values of sine and cosine for a few simple cases of right triangle. And there are actually only two simple cases of right triangles. Everything else is much more complicated. The first simple case is the right triangle was 45 degrees. This is 45 degrees, and this is 45 degrees. Well, what it means is, as you know, this is a square, actually. If I will just flip this particular triangle, because if I will flip it, it will be exactly the same triangle, which means it's also 45 degrees, and this is 45 degrees. So, it's 90, and this is 90. So, it's a rectangle, right? That's proven, right? But it's not just a rectangle, it's a rectangle with a diagonal being a bisector of angles, which means it's a rhombus, and rhombus and a rectangle is a square. So, that's what makes all these sides the same. So, both casualties are the same. And from the Pythagorean theorem, a square plus a square is equal to c square, which means c is equal to, this is 2a square, square root for both, a square root of 2, which means that sine of 45 degrees equals a over a square root of 2, right? 45 degrees opposite, two hypotenuse, which means 1 over square root of 2 or square root of 2 over 2. I just multiplied by square root of 2, both numerator and denominator, getting 2 in denominator and square root of 2 in numerator. All right, so, now, obviously, cosine of 45 degrees is exactly the same, this casualties is exactly the same, this casualties, so it's also a square root of 2 over 2. And, for the same reason, tangent of 45 degrees is equal to 1, because this casualties over this casualties, and they are the same, so the casualties over the casualties will be 1. So, these are three major function values for 45 degrees. Well, that basically is for this particular triangle. So, let's consider another triangle. Okay, another triangle is also very popular in geometry is 30 degrees, 60 degrees. Now, why is it popular? Because it's simple. Now, we were actually talking about this in one of the geometry lectures, but basically you understand that I will flip it over this side. This is also 30 degrees, right? I'll just flip it here, and this is 60. So, I have 60, 60, 60. It's the equi-angular, which is the same thing as equilateral triangle. So, all three sides are equal, but since this is equal to this, this A, and this is also A, and this is equal to this, so this is A, and this is 2A, sorry, A plus A, it's 2A. And this is 2A, right? So, in this right triangle's hypotenuse is twice as big as the smaller calculus, which lies across the 30 degree angle, which means that sine of 30 degree is equal to A over 2A, which is equal to 1 half. Now, and so is cosine of 60 degree, right? Which is, again, this over this. Sine of 60 degrees is equal to well, sine of 60 is opposite to 60 degree, which is this calculus, turns this one. Now, if this is A, and this is 2A, then, by the theorem of Pythagore, X squared plus A squared is equal to A squared. So, X squared plus A squared is equal to 4A squared, X squared is equal to A squared goes here, it's this. So, X is equal to A squared root of 3. So, this is A squared root of 3. So, this is A squared root of 3 over 2A, which is equal to square root of 3 over 2. And so is cosine of 30 degree. And obviously, tangent of 30 degree is equal to sine of 30 over cosine of 30, which is 1 half over square root of 3 over, so it's 1 over square root of 3, which is square root of 3 over 3. If I multiply by square root of 3 in both sides, and tangent of 60 is sine of 60, which is this over this, which is square root of 3. That's it. We have found all the values of cosines and cosines and tangents for 45 degrees and 30 degrees and 60 degrees. And that's basically it for right triangles. There is not much information which we can derive. This is a very simple case. And trigonometry of right triangles is really quite simple. Should you remember these values? Personally, I never remember anything like that. But I do remember that in case of 30 degree, for instance, I just keep in mind this particular drawing which signifies that the smaller of opposite to 30 degree acute angle is half of the hypotenuse. I do remember basically this drawing and this property of the 30, 60 right triangle and about 45 degrees right triangle. Obviously, again, you have to understand that the casualties is equal to another casualties and from there using the Pythagorean theorem everything else can be derived. So my personal preference is to derive all these formulas not to remember it. Well, quite frankly I've solved substantial number of different problems which are related and I actually do remember that sign of 30 degree is one half. But I don't really have to because I definitely remember I think and the fact that in 30 degrees case, casualties is smaller casualties is half of hypotenuse. That's enough. Everything else can be derived. Alright, so this is the end of this particular lecture. Try to derive yourself just using these drawings basically the values of trigonometric trigonometric functions for these major angles 30, 60, 45 and what I would also think recommend you to do is to derive not only these three major functions sine, cosine and tangent but also three others cotangent, second and cosecant. Just for practice and it's not big deal and then you can check on the internet or anywhere else actually that's it. Thank you very much and good luck.