 Hi, I'm Zor. Welcome to Inezor Education. This is the fifth out of six lectures in the mini series about how trinometric functions are calculated or get their values when the angle is one of the basic angles we really can calculate algebraically. So, this is about second, number five. We have still cos second. That would be number six lecture. So, secant and cosecant definitions. So, let's start with the secant today. By definition, this is one over cos sign of x. So, you know my preference not to memorize anything or this to memorize as little as possible. So, I will derive the values of secant for many different basic angles based on something very, very simple which I'm sure everybody remembers. And this is that in the right triangle if one of the angles is 30 degrees then the opposite calculus is one half of the hypotenuse. Now, my hypotenuse is equal to one in this particular case because I will apply it to unit circle. So, that's my unit circle. Now, obviously this calculus using the Pythagorean theorem square of this plus square of this is equal to square of hypotenuse. From here, x square is equal to three fourths and x is equal to square root of three over two. So, we know that. That's simple. No memorization. Everything is derived. That's square root of three over two. Now, another case also very simple is when it's 45 degrees. In this case, this angle is also 45 degrees. So, if this is hypotenuse equal to one then this is x and this is x x square plus x square equals to one x is equal to one half x square and x is equal to square root of two over two. So, that's another thing which we have to remember how to derive. So, these are the only things which you really should be very familiar with and I'm sure you are. So, everything else can be derived using the considerations of symmetry and properties of the functions like other, even, etc. So, let's start. The angle of two pi over three. So, let me divide each quadrant into three more or less equal pieces. So, this is 120 degrees which is two pi over three. Now, this angle is pi over three which is 60 degrees counting from the positive direction of the x-axis towards the vector to the point. Now, this is pi over six which is 30 degrees and in between you have pi over four which is 45 degrees. Now, using these properties this is one half because my radius is ak part of this which is in the unit circle equals to one. Now, this is also one half because if this is 60 degree this is 30 degrees, right? So, this is the calculus which is opposite to the angle 30 degrees. Now, this is square root of two over two and finally this is square root of three over two and this is also square root of two over two and this is square root of three over two. So, that's what we've got from these properties. Everything else we will try to derive using the considerations of symmetry as I said. So, 120 degrees this is an angle we are talking about and we have to find what's the secant of this particular angle. Now, obviously this is 90 degrees plus 30 degrees. Now, pi over three 60 degrees is 90 minus 30 degrees. So, these angles are symmetrical relative to the y axis from which follows that these points also are symmetrical relative to the y axis. Now, the coordinate of this point is x is equal to one half and y is equal to square root of three over two. Now, the cosine is the abscissa. So, one over cosine which is in this case is one over one half which is two. But let's calculate the coordinates of this point. Now, since these points are symmetrical and that was proved in one of the lectures where I introduced these basic angles. So, the symmetry means that they are on the same perpendicular to the y axis which means they are y-coordinate to the same but we are interested in x-coordinate which is abscissa. And the x-coordinate obviously is opposite in sign but the same in absolute value. So, this is one half and square root of three over two this is minus one half and square root of three over two. That's coordinates. Now, if you know the coordinates you obviously know sine and cosine of the function because the cosine is by definition abscissa and therefore one over cosine is one over minus one half. So, in this case I have the value of minus two. One over minus one half. Great. Next we have 135 degrees which is three pi over four. Exactly similar consideration. Now, the pi over four, 45 degrees is 90 degree minus 45. 135 is 90 degree plus 45. So, angles are symmetrical that's why the points are symmetrical and exactly the same situation. The ordinate is exactly the same and the abscissa are opposite in sign but the same in absolute value which means that for this particular I will have minus square root of two over two and square root of two over two. Now, the cosine is minus square root of two over two which means that one over cosine is minus two over square root of two that's square root of two. Next. Next is this which is 150 degrees or five pi over six radians. Now, 150 is 90 plus 60. 90 minus 60 is 30 degrees and you know the coordinates at this point. So, it's square root of three over two and one half. So, in this case I will have minus square root of three over two and one half and that's why the secant is inverse to minus square root of three over two which is two over square root of three you multiply by square root of three so it's this minus two square root of three divided by three. That's inverse square root of minus square root of three over two. Next, pi. Now, with a pi you don't really have to resort to symmetry or anything else. You know the coordinates at this point is minus one zero, right? The abscissa is minus one and the coordinate is zero and we need an inverse to a cosine inverse to the abscissa which is minus one. That's pi or 180 degree if you wish. Next, minus p over six. Alright. Yeah, I don't have to write about that. This is minus pi over six. Minus 30 degrees or minus pi over six. Now, you can use the symmetry relatively to the x-axis because this is plus 30. This is minus 30. It's just the direction but the angles are equal to each other. Which means that these points are symmetrical and they project into the same abscissa and the coordinates are opposite in sign but the same in absolute way. Well, we are interested in abscissa so the coordinate at this point would be one-half and the ordinate would be no, sorry, abscissa would be square root of three over two and the ordinate is in this case one-half and in this case is minus one-half. So, what's the secant of this function? That's inverse to square root of three over two and square root of three multiplied by square root of three numerator and denominator it would be two square root of three over three. Next, minus pi over four which bisects. It's minus 45 degrees minus pi over four and obviously it would be square root of two. I don't write it here it's abscissa is the same which means square root of two over two ordinate is minus square root of two over two. So, the secant is inverse to this which is two over the square root of two which is the square root of two. Next is minus 60 degrees which is minus pi over three. Obviously, symmetry is with pi over three. So, the same abscissa which is one-half and ordinate would be opposite in sign which is minus square root of three over two and inverse to abscissa would be two. That's the secant. Minus pi over two. Okay, now this point is easy because you already know that abscissa is zero and ordinate is minus one which means the secant is undefined because you cannot divide by zero. Next, minus two pi over three which is this. It's minus 120 degrees minus two pi over three. Obviously, symmetry with this point so this point has minus one-half square root of three over two. I retain abscissa which is minus one-half and they change the sign of ordinate which is minus square root of three over two. That's the coordinates. And we are interested in inverting abscissa which is minus two. Next is this which is minus one-thirty-five degrees or minus three pi over four. And this is symmetry with this guy. So I retain abscissa which is minus square root of two over two. Invert the sign of ordinate is also minus square root of two over two. So my secant is inverting minus square root of two over two which is square root of two is a minus sign. That's my secant. And next is this guy which is minus one-fifty degree which is minus five pi over six. Now, our symmetry is with this guy I know this coordinate so in this case I will have minus square root of three over two. I retain abscissa and change the ordinate and this should be inverted and the result is minus two square root of three divided by three. That's my secant. Well, we covered all the angles. Well, I specify minus five but that's the same as plus five. We go to the same point so it's the same secant. So, my point was that knowing the, this law of symmetry that if angles are the same are symmetrical relative to some vertical or horizontal axis then the position of the point is very easily calculatable from one to another. We just either change the sign of abscissa or change the sign of ordinate and knowing the coordinates of the three basic points where everything is from zero to 90 degree everything else can be calculated based on these considerations of symmetry and how to calculate these coordinates that's easy. You have these two basic triangles that's all you have to know all you have to remember everything else is derived with theorem or something like this. Alright so I do encourage you to go back to unisor.com take a look again at the notes to this lecture try to calculate all these values yourself using this unit circle and considerations of symmetry and check if you have the right answers I think it's a very good exercise. Good luck.