 Hi, I'm Zor. Welcome to Unizor Education. This is the second lecture of a series of lectures about how different trigonometric functions, what kind of values they take, if the argument is one of the basic angles which we were talking about before, like in the first quadrant it's 30 degrees, 45 degrees, 60 degrees, and then other quadrants symmetrically are manipulated as well. So this one, this lecture is about cosine and its values in different angles which I have chosen. It's not all of them, but most of the basic angles. And the methodology which I'm trying to use in all these cases is I don't remember the various of these functions, I'm calculating them based on something which I do know. And the only thing which I really know is that in the unit circle the angle of 30 degrees has an opposite catheters which is an originate of this point equal to one half of the k-partons. So the abscissa is calculated basically because the k-partons is one, this catheter is one-half, so this particular abscissa is calculated based on the Pythagorean theorem. So this is x, then x-square plus one-half-square which is one-quarter equals to one-square which is one. From here x is equal to square root of three over two. So my abscissa is square root of three over two and my originate is one-half. So that's this particular point, that's 30 degrees, that's pi over six, right? Now the next angle, basic angle is 45 degrees and the property of the 45 degrees angle is that the both catheters, both catheters are the same because this is 45 and this is 45 in the right triangle, right? So which means if this is x then x-square plus x-square is equal to one and the solution is x is equal to square root of two over two, right? So the coordinates of this point is square root of two over two comma square root of two over two, both abscissa and originate. And this is 45 degrees or pi over four. And the third major angle is 60 degrees and 60 degrees angle actually looks exactly like this triangle, looks exactly like this triangle. It has 30 degrees, 30 degrees and 60 degrees which is this, which means it's the same length of the catheter whatever was originate in this case would be originate in this case, so it's square root of three over two and whatever was abscissa in this case would be originate in this, which is, so the abscissa would be one-half and originate would be one square root of three over two and this is 60 degrees or pi over three. So that I know a little bit now and everything else I calculated based on the physics warrants here. Now from here I can find out the value of everything else just using the symmetry of these particular quadrants with this one. Now this lecture is about cosine, so I'm interested in the cosine, cosine is abscissa, so I'm interested in the first coordinate of these angles. Now my first angle is two pi over three, which is 120 degrees. So I have three points here. This is two pi over three. Now obviously since this is 120 it's 90 degrees plus 30. Now this point which represents 60 degrees is 90 degrees minus 30. So these angles are symmetrical relatively to this diameter which is y axis. Since the angles are symmetrical and according to the theory which I have proven when I was explaining what basic angles are in the definition of the very beginning of the trigonometry course, these points are symmetrical relatively to the y axis. Now symmetrical points relative to the y axis have obviously the same originate, but their abscissa which is this one are equal in absolute value but opposite in size. So if I'm interested in abscissa then one half should be reversed minus one half. That's abscissa and that's the value of the cosine for this particular angle. Next angle three pi over four which is this one, three pi over four. Now three pi over four is, now this is 120 degree and this is 135 degree. Well that's 90 degree plus 45, but 45 degree is 90 minus 45. So again these points are symmetrical relative to the y axis. That's why my abscissa in this particular point is equal to this one with a minus sign. So this would be minus square root of two over two. Now the last one in this quadrant is five pi over six which is 150 degree, five pi over six. And obviously 90 plus 60 will give you 150 degree, 90 minus 60 give you 30. So these points are symmetrical. So I take this abscissa with a minus sign, minus square root of three over two. Now the angle of pi which is this one is 180 degree. Now its abscissa is obviously minus one because we are in the unit circle so I don't have to do much. Now minus pi over six, minus pi over six is going this way. This is minus pi over six which is minus 30 degrees. Now we know that cosine is an even function. If argument changes the value from plus to minus with the same absolute value, then the function does not change the sign at all. So function retains the same value for x and for minus x. Okay which means that the cosine of this particular angle is exactly the same as the cosine of this particular angle. And that's obviously visible quite well from the drawing because they are projecting into the same point on the x axis which is an abscissa. And again from the symmetry theory which I was using this is plus 30 degree. This is minus 30 degree. So these angles are symmetrical and that's why these are symmetrical. They are the same perpendicular to the x axis projecting to the same point. And that's why their abscissa are exactly the same. So I'll just use the same square root of three over two. Now minus pi over four is this. Minus pi over four which is minus 45 degrees. Same thing. Symmetry is between this point and this point. 45 degrees. 45 degrees plus or 45 degrees minus. Points are symmetrical. Therefore the points are symmetrical. Therefore abscissa are exactly the same. So it's square root of two over two. Minus pi over three is this one. And the symmetry is with plus pi over three, right? And therefore abscissa is one half. So I will retain the same abscissa here. Minus pi over two is this one. It's minus 90 degree. Well, we don't have to do much thinking because this is definitely abscissa is equal to zero. Minus two pi over three is this one. And it's obviously symmetrical to two pi over three, right? So its abscissa is exactly the same. It's minus one half. Minus three pi over four. So this is minus two pi over three which is minus 120 degree. This is minus three pi over four which is minus 135 degree. It's symmetry with this guy. Now this guy has three pi over four has minus square root of two over two and that would be exactly the same here. We are projecting into the same point. And finally we have this one which is five pi over six minus five pi over six which is minus 150 degree. It's symmetrical to this guy and abscissa is exactly the same which is minus square root of three over two. Finally minus five is this one and abscissa is minus one. So methodology is very simple. Again, I remember only a couple of things like the categories which is opposite to a 30 degrees angle in the right triangle is half of the type options. Basically, everything else is derivable using the Pythagorean theorem and the considerations of symmetry. When points are symmetrical relative to the y-axis, then their ordinates are the same and their abscissa are opposite in signs but the same in absolute value. If symmetry is relative to the horizontal x-axis, then their abscissa is the same but the ordinates are opposite in sign but the same in absolute value. Well, that's it for major angles for cosine. Thanks very much and the next would be about tangent.