in this video we are going to look at the unit circle and some of the key trig ratios that we need in IB exams where we don't have a calculator so I've drawn up a few things here I'll explain as we go but firstly I start with two magic triangles these are these are two great triangles which I think you should commit to memory you may see these triangles in a tabular format in a table but the magic triangles are can be your best friend when it comes to this topic and what they are used for is they help us find sine cos and ten of Te angles now I've drawn white my two triangles here and I'll quickly explain them this is a triangle where we have a 45 degrees and a 45 degrees now in radians that's it's PI on for because it's a quarter of 180 and 180 is PI so we have degrees angles on the inside and our radian angles on the outside and what this is telling us is that if we do sine of 45 or sine of pi and 4 if you're doing radians you just need to go opposite over hypotenuse which is sine and you'll get the value so sine of 45 would just be opposite 1 over the hypotenuse root 2 and we can verify this on our calculator but we need to be able to solve sine of 45 without a calculator in IB exams okay now if we need cause of let's say 30 degrees we can go to the triangle that has 30 degrees which is this one here and the Radian of 30 degrees is part 6 because 30 is 1/6 of 180 so that's PI okay so cos is adjacent over hypotenuse so you would take the adjacent which is root 3 over the hypotenuse and we get the exact value for root 3 onto which we could verify on our calculator okay and if we want to for example do 10 of let's say 60 degrees or PI on 3 radians we can go to the triangle that has pylon 3 which is this down here and take 10 and 10 is opposite over adjacent it'd be root 3 on one which we can just say is root 3 so these two triangles help us to find all of the key trig ratios that we need in non calculator exams we can find sine causing 10 of 30 45 and 60 degrees now how does this relate to the unit circle well I've drawn a unit circle over here and we know that the unit circle starts at the origin in the middle and it has a distance of one unit out to our circle points so for example if we go all the way over to this point here you'll have one in the X and zero and the wire so that's why it has the coordinate 1 0 so on so on now each of these t top and top bottom and side the side points has degrees in Radian value so we start here and we go around get to 90 which is part two then we'll get to 180 which is PI and then 270 which is three part two and then back to two pi or 360 degrees okay so if I were to draw some angle let's say 30 degrees I'm going to go up to about 30 and if we work our way down and say this is 30 degrees what we know is we naturally know the value of sine of 30 when we can go to where 30 years and sine of 30 would just be take 30 opposite over hypotenuse one over two and what what that actually means down here sine of 30 is one over two which I'm going to rise a decimal zero point five just so we can kind of see what that means and what that means is if we have a total length of one going up as you can see here the height of this line will just be 0.5 and that just means that we're going to have a positive value here where the height is one-half of the total length of the line because the total length was one so that's what sign a clean sign is the height component of any angle compared to its full length the hypotenuse so that's what I will get about you of 0.5 now if we did cause of 30 cause of 30 or we can go to our magic triangle undo cause which is root 3 on 2 and root 3 on 2 as a decimal is around somewhere between 0.8 and 0.9 I think it's about 0.85 ish and that kind of makes sense now because this angle angles quite shallow the X component of this diagonal line will be 0.85 or 85% of the total length so hopefully what you're seeing here is that sine is the highest component and cos is the width component of any diagonal line now what we can then do is we can do sine cos and tan of these key angles 30 45 and 60 in different quadrants so if I chose to do the angle over here which we go around this will be a hundred and fifty degrees we can think about what this angle must be that's 150 and this is 180 this must be 30 so if I took sine of 150 I'm just funning the heart here at 150 which looks to be the same height at 30 so that's why sine of 150 will also be positive 1/2 or 0.5 but if I did sine of 210 which is down here notice that I will have the same the same value of the heart 0.5 but now it's negative because it's going down so sign up to 10 is negative 1/2 or negative 0.5 and sine of this one here would be 30 down to 330 this would be the same height but it would be negative again so sine of 330 would be negative 0.5 and you can do this for cos Cosby in the width and tan but all of our key angles and what what we can actually do is a bit of a shortcut you may have seen these letters before a in the first quadrant s and the second quadrant T in the third quadrant and C in the fourth quadrant these stand for all I stands for all and that means if you take sine cos and ten of any angle in this first quadrant it will be positive because the heart will be positive that width will be positive and ten is that actually the opposite over the adjacent it's the Y over the X so they'll all be positive in the second quadrant here only sign that's why it's the S quadrant is positive because any angle in the second quadrant will have a positive heart but it has a negative width it's going left of the y-axis so only signs positive in the third quadrant only tan is positive because both sine and cos the height and the width will be negative and ten is actually negative over a negative which will turn it positive and then in the fourth quadrant only causes positive because it's going down the height is negative so sine will be negative and tan will be negative only cause the width will be positive so this is the unit circle it's a it's a very tricky topic to sort of pick your head around but the goal of this video is to firstly if you can solve sine cos and ten of 30 45 and 60 hopefully then you can see the big picture and how it relates to a full circle where you can then solve sine cos and ten of 30 45 60 in any quadrant just using the positive and negative values depending on what quadrant it is in okay I encourage you to practice a bunch of questions on this topic and good luck