>> We start with an equilateral triangle, that's and angel, excuse me, that's a triangle with all angles the same measure, the same opening and every one of them is 60 degrees. So, it's 60 degrees here, 60 here and 60 there. Now, in order to create right triangle situation so that we can study our ratios, we do this. We construct a perpendicular bisector from this angle down to this side. So we do this. Now if this is, if it's perpendicular, then we have a right angle here and if it's a bisector up here, then this is a 30-degree angle. All right. now, we have a, we have kind of a neat situation here and now notice that because this is a bisector, that whatever length side this is, that this distance is just half of that. You see? This is the same as that and we're taking half of it. All right? now that's really important when we go through the business of just choosing values for the lengths of sides all right, and we're going to do that right away here. That if I want to make the short side as convenient as I possibly can, then I would make it of length 1 and if this 1, then that's also 1 and the whole length here is 2 and therefore that's 2. So this length is 2. Okay? So now, we have a couple of sides labeled and we can generate the third one by using the Pythagorean relationship. I'll just take off this part of the figure and away we go. Now this one, is one where we're finding the length of one of the legs. We know the hypotenuse is 2 but one of the legs is unknown to us. So, let's see, we have C squared is 2 squared, is equal to, now I'll let A be the one here, and we're finding B. So, we have 4 minus 1, this becomes 1, subtracted on both sides, we have this. B squared is 4 minus 1 or 3. B then is the square root of 3. Okay, so this side is the square root of 3 and away we go with our trig ratios. Now again though, before we even get to the ratios, this is your basis for remembering the trig ratios. All right? This is our reference for those ratios. And this is often called a reference triangle for that reason. And no just notice that it is 1, 2 the square root of 3. 1, 2 the square root of 3 and the one side is opposite the 30-degree angle. All right? It's the short side. The short side of a triangle is opposite the smallest angle always. And so no wonder this is the case here for this particular triangle. All right, now let's generate our ratios. We have, let's see, from the perspective of the 60 degree angle, and we're going to do both of them here for this little reference triangle, but for the 60 degree angle, the sine is the square root of 3 over 2, opposite over hypotenuse. Sine of 60 degrees, square root of 3 over 2. Cosine of 60 degrees, that's 1 over 2. 1 over 2, 1/2. The tangent of 60 degrees, square root of 3 over 1, square root of 3. The others are reciprocals of the ones above. Now, we can, we can emphasize the idea rationalizing but I'm not going to mess with it too much right now. I just want to emphasize though, that these are reciprocals and, therefore, the co-tangent of 60 degrees is 1 over the square root of 3 which can be rationalized and then secant 60 degrees is the reciprocal of cosine 60 degrees and the reciprocal of 1/2 is 2 over 1 or simply 2. The cosecant of 60 degrees, let's see, and the reciprocal of sine would be flip this over and we have 2 over the square root of 3.