 Hi, I'm Zor. Welcome to InDesert Education. I would like to talk about some special angles which trigonometry likes to deal with. Why are they special? Well, because they are easy to deal with. Now, you know for instance from the geometry such a great quality of the right triangle with a 30 degree acute angle and 60 degree another that this particular catheters is equal to half of the hypotenuse for instance which is very easy to prove just with another triangle like that that would be also 30. So it's 60 together and this is 60 as well so this is equilateral triangle and since this is half of this side so it's half of this side. So these properties are basically making triangles which is this one like 30, 60 and also 45, 45 right triangle special in a way that it's easy to deal with these. In trigonometry we are talking about ratios between catheters and hypotenuse or one catheter and another catheter and since everything is so easily related to each other the values of trigonometric functions of these basic angles like 30, 45 and 60 are very easy to calculate and deal with. So considering we are dealing with angles not only within this range from 0 to 94 right triangles but for an entire range entire cycle of 360 degrees or 2 pi in regions there are some other angles which are also special in the same sense it's easy to calculate the various of all trigonometric functions and that's what I'm going to talk about in this lecture. So let's go back to original definition of the trigonometric functions as ordinates and abscesses of different points on the unit circle or the ratio among them. So we will devote our attention first to the first quadrant and the three important angles are 30 degrees, 45 degrees and 60 degrees. Now I will also talk about 0 degree when the point A actually is coinciding with the direction of abscesses and 90 degree when the point is coinciding with the direction of the y-axis. And then from these angles and their degrees in there and the coordinates of the point A we will derive all other everywhere. What I would like actually to say before that that the same picture which is in the first quadrant it should actually be exactly transformed into every other quadrant. So we have this is one-third basically 30 is one-third of 90, 45 is half of the 90 degree and 60 is two-thirds, right? So the same three angles would be here. But since we are measuring the angle from the positive direction of the x-axis counter-clockwise so there is another measure obviously it's not 30 degrees, 120 degrees this is 135, etc. I'm not going to write it right now but you have to understand that each quadrant has exactly the same arrangement the same three vectors are pointing to three points which constitute these major angles one is in the middle and the other two are cutting one-third. So together with the top and left and right and top and bottom points they constitute how many different special angles? 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 So we have 16 special points Now each angle which is characterized by a point and angle again is calculated from here counter-clockwise as a positive angle or clockwise as a negative angle Now for each of these angles there are six values of different trigonometric functions like sine, cosine, tangent, cotangent, secant and cosecant So 16 different points times six different functions you have 96 different values which you might be asked to remember obviously nobody remembers it What's important is to understand the system how they are formed and that's what we are going to talk about Basically derived whenever you need any value of any angle using this system Basically my personal recommendation is all you have to remember is that 30 degree is characterized by a vector of opposite calculus of one-half of the hypotenuse Everything else can be derived and for 45 degrees they also can derive everything because you know that the 45 degrees triangle has the same catechism the same catechism since they are equal to each other This is 45 degrees and this is 45 degrees So everything is derived using Pythagorean theorem and some symmetry properties which we are talking about Don't memorize anything Again the only thing you might actually have to memorize and it's kind of almost automatic that the 30 degree triangle right triangle has an opposite catechism equal to half of the hypotenuse Everything else is derived Alright so Now we are ready to approach our problem and let's talk about the first quadrant only So we have 30, 45 and 60 degrees Now as I was already saying we can derive everything but what you really need is co-ordinates of this point Why? Obviously because for instance sine of this is an originate cosine of this is an abscissa of this triangle of this point actually and the tangent is ratio between ordinate and abscissa cotangent is the reverse abscissa over ordinate second is 1 over cosine which is 1 over abscissa and cosecant is 1 over sine which is 1 over ordinate So if you know basically one particular segment and you know it, it's one half of the hypotenuse everything else is basically derived from the definition So let's talk about these angles So co-ordinates of point A Now, since you know that the radius is always 1 and I was already talking about the opposite cateches to the hypotenuse is 30 degrees equal to half of the hypotenuse so we know that the ordinate is equal to 1 half Now what is another catechus which is abscissa of the point A? Well, we know the theorem the Pythagorean theorem, right? to the square of the hypotenuse 1 square is equal to square of this and you know what this is Well, let's just solve it I mean, it's not big deal It's 1 equals to x square plus 1 quarter from this x square equals to 3 quarters and x is equal to square root of 3 over 2 So that's the co-ordinates of the point A when it represents the angle of 30 degrees Now, let's move on 45 degrees I will not use the letter A anymore I'll just use the co-ordinates Now, if this is the 45 degrees so you know that this is the right triangle this is 45 and this is 45 obviously so the catechus one catechus is equal to another catechus, right? So if this is x so it's x square plus x square equals to 1 square because the catechus is equal to 1 that's the radius Well, from this to x square equals to 1 x square is equal to 1 half x is equal to 1 over square root of 2 which is the same as square root of 2 divided by 2 I just multiply it by square root of 2 both as you understand I'm not really losing any roots of this equation because x represents the length which is positive so that's why I'm taking only the positive value So in this particular case both abscissa and coordinate are equal to square root of 2 over 2 Finally, we have a 60 degrees but now 60 degrees is easy because if this angle is 60 degrees then the opposite angle is 30 degrees, right? These are right triangles if one angle is 16 others if your angle is 30 so this catechus is equal to half of the hypotenuse which is 1 half So we have basically the same triangle as in this particular case they're just positioned differently but it's the same other catechus can be calculated like exactly the same calculation as before So in this particular case our abscissa is equal to 1 half and that's why the coordinate we have already found using these equations and Pythagorean theorem basically is equal to square root of 3 divided by 2 Now in this case when the angle is equal to 0 well the coordinate of this point is abscissa is equal to radius which is 1 and the coordinate is equal to 0 obviously and the opposite side not the opposite 90 degree side that's when abscissa is equal to 0 and the coordinate is equal to the radius which is 1 So we have basically covered all the points in the first quadrant all the special angles or basic angles Now we found their coordinates the coordinates of the points which represent these angles Now how can I find any trigonometric function? Well, what's the definition? The definition of sin is ordinate, right? So the sin of 0 is 0 sin of 30 is 1 half sin of 45 is square root of 2 over 2 and sin of 60 is square root of 3 over 2 and sin of 90 is 1 Now the cosine cosine is abscissa So for 0 is 1 for 30 degrees is square root of 3 over 2 for 45 degrees is square root of 2 over 2 for 60 degrees abscissa is 1 half the cosine and for 90 degree cosine is 0 What else? Let's say we want the cosecant is 1 over sin sin is ordinate So for this the denominator is equal to 0 so it's not defined for this it's 2 over 1, right? So it's 2 for 45 degrees it's 2 divided by square root of 2 which is square root of 2 for 60 degrees it's 2 divided by square root of 3 or you can change it it would be 2 divided by square root of 3 the same thing is 2 square root of 3 divided by 3, right? It's customary to have radicals on the on the numerator side so I multiply by square root of 3 both sides So in any case any trigonometric function can be derived from these So I'm not going to remember anything the only thing which I really have to memorize is Pythagorean theorem and one particular property about 1 half for 30 degrees everything else is derivable and that's how I suggest you to basically approach these particular things Don't memorize anything except the way how you can derive the whatever is necessary Well obviously we have to memorize the definition of the cosecant that this is 1 over sine or tangent is sine over cosine, etc By the way tangent of 45 degrees for instance it's sine over cosine sine is ordinate, cosine is abscissa so it's square root of 2 divided by square root of 2 it's 1, well tangent of 45 degrees is 1, that's it Alright, fine Now we have to move to other quadrants and before doing that I would like to prove a very very small theorem which will be very handy It's about symmetry You see these things are symmetrical these three are symmetrical to these three relative to vertical axis Now these three are symmetrical to these three or these are symmetrical to these three relative to horizontal axis So I will use this symmetry to derive the various of these guys How? Very simple Let's consider you have a circle you have a diameter and you have let's say it's s t and that's and you have two points p and q such that these two angles are acute congruent to each other which basically means that angles are symmetrical relative to the diameter Then my point is that p and q are symmetrical as well Now what does it mean that they are symmetrical when are the two points symmetrical to each other relative to a line when they are on the common perpendicular and on the same distance from the intersection of this perpendicular with a line, right? We have to prove that p and q are on the same perpendicular and on the same distance from the diameter Alright, let's prove Let's drop perpendicular from p and from q I'm not sure how visible it is but I deliberately left little distance between them because in theory they might not coincide If they coincide these two perpendicular then I know they are on the same perpendicular that is the same one line but maybe they are not I don't know, right? So let's define the point i and the point j as the points where perpendicular start falling to s t Now consider since these are perpendicular, these two triangles O, P, I and O, Q, J are right triangles, right? Now, O, P and O, Q are radius, so they are equal they are hypotenuses for these triangles and angles we know by the condition of the theorem they are congruent Now, obviously right triangles are congruent by hypotenuse and in the Q-tangle so if they are congruent that means that O, I equals to O, J so I and J are on the same distance from O which means they coincide it's the same point actually also it means that P, I and Q, J are also the same lengths which means not only P and Q but also are on the same distance from the base of these perpendicular which proves that they are symmetrical Now, it's quite obvious theorem and to prove it was just, you know, basically use one particular theorem about congruence of right triangles because it basically trivial theorem and also not only it's trivial to prove but you actually see that this is the right thing I mean, if angles are symmetrical and this is a circle this is a diameter diameter is also kind of symmetrical everything is symmetrical so P and Q must be symmetrical that's important so we have proven this theorem that if angles around the diameter are symmetrical then the points of intersection of these angles with the circle are symmetrical as well now let's take a look at this what is this angle? that's one third of the game it's from the 90 degree we just took only 30 so this is 90 plus 30 it's 120 so we're talking about this point which is 120 degree it represents the angle of 120 degree but now consider this point and this point which is 60 degree now this point is 120 which is 90 plus 30 this is 60 which is 90 minus 30 so this is also 30 so angles are symmetrical well if angles are symmetrical then the points are symmetrical they are symmetrical to the y-axis now if points are symmetrical relative to the y-axis it means they project onto the same ordinate so ordinate is the same now since they are symmetrical then this distance is equal to this distance and these are abscissists but they are on different sides so if this has coordinate uv then this point should have coordinate minus uv because abscissa is negative and this abscissa is positive but they are equal in absolute value but the ordinate is the same so v and v from this we immediately get the coordinate of this minus 1 half square root of 3 over 2 I just took this and use this principle of symmetry and got this how about this guy this is 45 here and this is 135 which is 90 plus 45 and relative to 90 minus 45 which is 45 so these points are symmetrical which means this is minus square root of 2 over 2 and square root of 2 over 2 and the third point this is 90 plus 60 which is 150 degree 150 degree and it's symmetrical to this one to 30 because this is 90 plus 60 this is 90 minus 60 these angles are symmetrical so the points are symmetrical which means we have minus square root of 3 over 2 comma 1 easy so everything is symmetrical now these guys are also symmetrical so this is 180 and by the way in this particular case if the point is here it's 180 so coordinates are minus 1,0 minus 1 is and 0 is now but this is 180 plus 30 which is 210 degrees this is 180 plus 45 which is what? 225 and this is 60 which is 240 alright now they are symmetrical relatively to the x axis the diameter right so whatever angle goes here or angle goes there if angles are the same then points are relatively to this so the 210 point is symmetrical to 150 this is 180 minus 30 this is 180 plus 30 well if points are symmetrical relatively to the x axis then their abscissas are the same they are projecting to the same point that's abscissa but the ordinate which is projection to this they are equal in the magnitude but opposite in sum so we will have this which is immediately following that this is equal to minus 1 square root of 3 over 2 comma minus 1 half now this 225 is symmetrical to this this is 180 plus 45 this is 180 minus 45 which is 135 and therefore I will have the square root of 2 minus divided by 2 with a minus sign and another is minus sign 240 is symmetrical to this this is 180 plus 60 180 minus 60 is 120 so here we have minus 1 half minus square root of 3 over 2 finally if point is here that's 270 degrees the coordinate of this point is abscissa is 0, ordinate is minus 1 next we can actually compare these gaps so this is 30 degrees so this is 300 right so this is 300 this is 270 plus 45 which is 315 this is 3 330 alright now 300 is 360 which is the whole circle minus 60 and it's symmetrical to 60 right this is 60 degree and this is 60 degree 0 and 360 is exactly the same thing so it's 0 plus 360 360 minus 60 it's exactly the same alright so this is symmetry relatively to the x axis which means again we are changing the ordinate so for 300 we get 300 degrees we have 1 half minus square root of 3 over 2 for 315 we get this square root of 2 over 2 minus square root of 2 over 2 and for 330 we have square root of 3 over 2 minus minus 1 so we have coordinates of every special point on our circle which represents every special angle now by the way it's the same to say 330 or to say minus 30 it's exactly the same point sine or cosine or anything else and the coordinates will be exactly the same so it doesn't really matter whether I'm using this notation or minus 30 minus 45 minus 60 minus 90 etc it doesn't matter so in any case we have coordinates of all special points and if we have coordinates we can construct from this any function so again when we repeat it from the beginning what I am suggesting is to remember only this 1 half everything else is very easily derived using symmetry you just have to always imagine yourself this unit circle and the theorem that whenever you are symmetrical relatively to some diameter then your points will be symmetrical as well and now symmetry relatively to the y axis means that your coordinate is the same and abscissa are different in sine but the same in absolute value if you are symmetrical relatively to the x axis then your abscissa is the same but coordinate is opposite in sines and the same in absolute value so from these numbers as I have written it here which you do not have to memorize you can derive the value of any function as long as you know the definition of the function sine is ordinate cosine is abscissa tangent is ordinate divided by abscissa cotangent is abscissa divided by ordinate what else secant is 1 over cosine which is 1 over abscissa what else secant is 1 over sine which is 1 over this that's it I have covered all these special values and I will use these special values to basically present different problems related to each and every function now when I will talk about functions I will present certain calculation problems like calculate the value of cosine of 150 degrees and you have to really you know based on this imagine what is 150 degrees 150 is 180 minus 30 so it's symmetrical to this or 150 is symmetrical to 90 minus 60 which is basically 30 so there is a symmetry so you have to find which symmetry is typical 2 which symmetry to use and then correspondingly change the value of whatever the basic value which are actually 3 values 1 half square root of 2 over 2 and square root of 3 over 2 everything else is again interoperable and that's how you can get the value of coordinate of the 150 degree point and from the coordinate you can find out everything all the functions by manipulating the assistant audience okay that's it for today thank you very much and try to mentally go through this circle again just by yourself to basically feel this symmetry and feel that you don't really have to remember everything you have to remember basically some very very few items information and everything else can be derived thanks very much good luck