 Hi, I'm Zor. Welcome to Unisor Education. I would like to continue talking about trigonometric functions and primarily their values for some basic angles. The lecture which preceded this one was about sine. What's the value of sine for major or basic angles where we can easily calculate its value? Now we will talk about cosines. So what are the angles we're talking about? Well, it's zero radians, which is zero degrees. It's p over six radians, which is 30 degrees. It's p over four, which is 45 degrees. It's p over 360 degrees. And p over two. Actually, sometimes I'm saying p sometimes pi depending on which language I have in mind, Greek or Latin or English, whatever. So for any of these angles, we will calculate the value of cosine. Let's go back to the basics. This is our unit circle. These are crossing with axis. This is our point A with coordinates p, q. And this is an angle which we are talking about. Well, it's really following exactly the same pattern as the previous lecture for sine, except instead of coordinate, I have to consider axis. So this is q, this is p, this is capital P. This is capital Q. So, same drawing. And I will try to be as fast as possible because it's really easy and again, resembles the previous lecture. For angle of zero, A coincides with x, and obstesis is equal to one. So cosine of zero is equal to one. Now, if this is p over six or 30 degrees, then the opposite calculus, which is q, is equal to half of the k-paginus, which is one because it's a unit circle. So by Pythagorean theorem, p square is equal to hypotenuse square, which is one, minus this calculus square, which is one quarter, which is three quarter. So the letter p would be equal to square root of three over two. So the cosine of pi over six equals square root of three over two. Now, what if this is 45 degrees, p over four or pi over four? English p, Greek pi, whatever. Anyway, pi over four, so this angle is also pi over four, 45 degrees. So this is not only the right, but also equal at root triangle, which means that p and q are the same. And again, by the Pythagorean theorem, p square plus q square is equal to one. Now, instead of q square, we can put p square, which is the same thing as we know. So it's two p square equals to one, times two for a good square root. Now square root, principle variable, so it's two p equals to square root of two. So p is equal to square root of two over two. Cosine of p over four equals square root of two over two. Next, 60 degrees, pi over three. Well, if this is 60 degrees, this is 30 degrees, pi over four. So the opposite to 30 degrees sides of this right triangle, which is actually our ccp, is equal to capital capital x. So there is nothing to talk about here. We immediately have it one half. Finally, if it's pi over two, which is 90 degrees, a coincides with y, and it subsists as equal to zero. Now, thinking about other quadrants of the coordinate plane, again, very analogously to whatever I was doing with the sine. Cosine has properties of being an even function, which means that's an even function, which means for every angle in this particular quadrant, we can have the corresponding symmetrical angle in the positive quadrant, and it will be the cosine, which is obsessed, will be the same. Now, if we are talking about this quadrant, then we can have a centrally symmetrical point on that side. So if this is x, this is pi plus x, right? So we know how to calculate this, and to calculate cosine of pi plus x, we know this is equal to minus cosine of x. That's another property which we discussed when we were talking about cosines. And again, it's kind of obvious because since these are symmetrical, centrally symmetrical points, this obsessed is positive, and this is negative or y-scores. And finally, this is in this quadrant, we can consider a center, not centrally, but a reflection relative to the y-axis, and consider this angle instead of this angle. So we have to really, from pi, subtract this. So we need this. And the ordinate, again, would be negative in this case relative to this one. So this is an obvious equality. So these are properties which we discussed before I just briefly mentioned them, and they can help you to calculate any other value of major angle. Let's say you have an angle of 135 degrees, which is 180 minus 45. 180, which is pi minus 45. You know, it's minus cosine of 45 degrees, which is pi over 4, which is minus square root of 202, et cetera. That's it. Very basic, very short lecture. And we'll continue talking about major angles for tangent, cotangent, et cetera. Very small, very brief lectures. That's it for now. Thank you.