[Music] oh [Music] okay welcome to this video we are where we are looking at all of the formulas that you need for the higher tier paper now in the previous video we looked at all of the formulas that you need that appear on the foundation and on the higher paper so you do need to make sure that you check out that video as well as that video covers all of the rest of the formulas that you still need on the higher paper but this video is going to look exclusively at those that you only need if you are sitting the higher paper so if you are setting the foundation paper go into the description and look at the other video part one looking at all the formulas that you need to pass your gcse exam now if we quickly look at that video so you can see how it works you can see that within the video all of the links are in the description so you can actually click onto the video and if you click the little link in the bottom left it bookmarks everything for you so you can scroll through the video and you can have a look at any of the videos or formulas that you want to have a look at you can then click back in the into the description and all of those videos will be linked for you so that you can go and watch the full lesson so hopefully that's going to be really useful for you but let's go back to our video so as i said this is going to be the higher only paper formulas and with that being said i'm the gcse math tutor and let's have a look at our first formula so the first formula that we're actually going to have a look at is going to be following on from the previous video and that is going to be the volume of a pyramid now the volume of a pyramid is equal to 1 3 times the area of the base multiplied by the height so if we look at an actual pyramid we first need to think about the actual base so the base here and we're going to put some numbers in as we did on the previous video we're going to see say that the lengths of the base are six and six now we also need to know the perpendicular height so if we also put a perpendicular height in and that is going from the peak of the pyramid down to the very center of the square base and we're going to say that that is eight so the formula here we would do 1 3 multiplied by the area of the base here is the square so we would do 6 times 6 and then we would multiply that by the height of 8 and that would look like this 6 times 6 you could simplify to 36 if you want although if you have a calculator you can type that all in straight away if you don't a third of 36 is 12 so we'd have to do 12 times 8 and 12 times 8 comes out as 96 and this is a volume so we would say centimeters cubed so that is our first formula let's have a look at our next one now the next one we're going to have a look at is the quadratic equation the quadratic equation is the nastiest looking out of all of them in my opinion but it is x is equal to negative b plus and minus the square root of b squared minus 4ac all over 2a and this is applicable to when we have a quadratic that is in this form ax squared plus bx plus c and when it is equal to zero so when we are looking at a question like this on a basic level let's just have a look at one where it says solve the equation and the equation is three x squared plus five x minus four is equal to zero and we need to give our answer to two decimal places so for this here the first thing we look at is our values of a b and c and they are the numbers highlighted in that particular order so for this one and we would write them down a is 3 b is positive 5 and c is negative 4. and when we put that into our formula we just need to be careful with some of those numbers so as we can see in the actual formula b we have minus b to start with so the b value the sim the sign in front of it is going to change and we just need to be careful about subbing some of them in so as you can see in the way that i've subbed them in i've also put the five squared in brackets there and that is just to make sure in case b is negative we don't get any errors on the calculator and also i've put the negative 4 in brackets at the end and i always do that when i'm substituting i always put the negative number in a bracket now if you want to see more on this obviously you can check out the full video on the quadratic equation but going through this relatively quickly we would type that into our calculator once using the plus sign which gives us this answer and once using the negative sign there in front of the square root which gives us this answer now this actual question does say to give your answer to two decimal places so we would round the first one to 0.59 and we would round the second one there to negative 2.26 now they are our final answers we would give both of them and for the majority of questions using the quadratic equation you are going to get two answers so that's the quadratic equation let's have a look at our next one so now we're going to have a look at the sine rule so the sine rule is a over sine a which is equal to b over sine b which is equal to c over sine c and we use this when looking at triangles now specifically you would use this when looking at a non-right angled triangle although it can actually be used with right angle triangles as well but we already know from the previous video we can use socatoa or sin coz and tan for looking at our right angle triangles so if we throw some lengths into this triangle and think about why we would use sine with this particular triangle now when you are looking at them you want to identify pairs of opposites so what i mean by that is that opposite to 30 degrees we have a length that we're looking for and we can call that a pair of opposites and we can also look at the length that is opposite 62 and we can see that we have two pairs of opposites in this question so that means we can use the sign rule so what we need to do is label the sides now what i always do is i label a and b to start with so let's just label this angle a and therefore its opposite side would be little a we'll also label the 62b and therefore the opposite side is little b and because of this process we only ever actually need to use two of the pieces in our formula so i'm never really going to use the equal c over sine c although that doesn't mean that we shouldn't label the sides as our other two sides will be big c and little c now if we put the values that we actually need into our equation so the little a and the big a and the little b and the big b into our fractions there we end up with something that looks like this now as you can see in that formula we have got x on the top of the left hand fraction and we want it to say x equals so we need to multiply both sides by sine 30 and if we multiply both sides by sine 30 the sine 30 would move on to the numerator on the right hand side and again if you're unsure on any of this do check out the full video on the sine rule when you plug that into the calculator we get the answer 2.831 and some extra decimals there but if we round this to one decimal place for the purpose of our question we would get the answer 2.8 centimeters now this is the sine rule for working out lengths if you are going to work out an angle there is a different formula where you flip the fractions over so instead of it being a over sine a you have sine a over a and the same for obviously b and c so again if you're unsure on how to approach that then do check out the full video where i go over this in depth but with that being said let's have a look at our next formula so logically our next formula here is going to be the cosine rule and the cosine rule is shown on the screen and again it's quite a long one we have a squared equals b squared plus c squared minus 2bc cos a and when we are looking at the cosine rule again we are looking at a triangle and again this is for looking at non-right angle triangles but how does it differ from the sine rule well if we put some lengths in and have a look at this again we'll have a look for those pairs of opposites so we can already see that opposite the eight centimeters or the six centimeters there is no angle so if we look at this one to start with opposite six there is no angle so we do not have a pair of opposites but we cannot can see the opposite the 60 there is the length that we're looking for and we can include that as a pair of opposites but unfortunately the final one is not a pair of opposites we have nothing opposite the eight so this is our indication that we're going to use the cosine rule now when i'm doing this i always stick to using one version of the cosine rule where we have cos a at the end and because of that i always label the angle that i'm going to use in the question my big a and then the opposite side little a otherwise if you don't do this you could end up having to learn three versions of the formula which i wouldn't recommend but is also a possibility so if we start to label the other sides then and you can put these in either order but let's just call those b and let's call this angle c and the opposite length there little c and then we can actually just plug that straight into our formula so putting all of the numbers in place we end up with something that looks like this so we have x squared is equal to 6 squared plus 8 squared which is our b and c take away 2 times 6 times 8 cos 60. now this is an interesting one because cos 60 is one of our exact trigonometric values and is equal to one-half and again it's not necessarily one that you have to remember because you can actually work these out and you can check that out in obviously one of my videos on working out the exact trig values but because cos 60 is equal to a half when we put all these values into our calculator we get quite a nice number we get 52. unfortunately though the square root of 52 is not a nice number and we do get a long decimal which again i'm going to round to one decimal place so we would say our answer here is 7.2 centimeters and that is how you would use the cosine rule now again when it comes to the cosine rule you can rearrange it in order to look for an angle and the rearranged aversion here is that cos a is equal to b squared plus c squared minus a squared all over 2 bc and when you are working out an angle you have to use the inverse of cos just like when you are doing any other versions of trigonometry again that can be seen in the full video on the cosine rule that i've got linked in the description so let's have a look at our next formula which is going to be the area of a triangle specifically when we are using sine and that is going to be half a b sine c so we use this formula when we are unable to work out the height of the triangle so if we throw some lengths into here we can just say that the one angle is 30 and the other two there are seven and eight now other angles and lengths could be given to you here but i've made it the most basic question possible just to show you how to use the formula now you can see in the formula that the big c we are going to be using in this formula so the angle that we're going to use i'm going to label as c so opposite that is little c and the other two can be labeled in any order so let's just put a and b in either order now you just need to put these values into your calculator so writing down the formula would give you one half times seven times eight times sine thirty and again this is going to be an interesting question because it involves sine thirty which is one of our other exact trig values again this comes out as one half so cos 60 and sine 30 are our values that are given that are one-half although some of the others are slightly more complex but here when we type that into our calculator we get the answer 14. again that is the area so we would give that in centimeters squared and that would be our final answer for this question 14 centimeters squared now technically that is all of the formulas that you need to remember but there are some bonus formulas here that i thought was really important if you are sitting the higher tier paper and the first of those is the area of a sector and this follows on nicely from the last video looking at the area of a circle as it is pi r squared but we multiply it by the fraction of the circle we're looking at and that has given us theta over 360 where theta is the angle in the sector so if we have a look at an example of a sector and we put in obviously we're going to need a radius which this time is the length shown on the screen so if we put the angle and the radius in we'll say it's 62 degrees with a radius of 8 we just put that into our formula which would be pi times 8 squared multiplied by and the angle there is 62 so 62 over 360. you can type that straight into a calculator and it gives you 34.627 but if we round that to let's say one decimal place again we get an answer of 34.6 again it's an area so that would be centimeter squared now that is for working out the area of a sector but you could also have to work out the arc length and if you have to work out the arc length it follows on nicely from circumference as it's pi times diameter again multiplied by the fraction of the circle that you're looking at again if you're not sure on this topic i will link the full video in the description for you moving on to our next formula we're going to look at direct proportion so with direct proportion you have a formula which is x is equal to k y to the power of n now when looking at direct proportion any letters could be used in the formula so this is where x is directly proportional to y to the power of n so an example of this would be if a question said x is directly proportional to y squared when x is 18 y is equal to 6 and we want to find out the value of x when y is equal to 8. so we would say that x is directly proportional or equal to y squared or ky squared and we're going to find the constant of proportion here so we sub in those values 18 and 6 and we get 18 is equal to k times 6 squared then you can divide by 6 squared on both sides and you get the answer k is equal to a half now we can put a half back into our formula so we would have that an x is equal to a half y squared rather than x is equal to k y squared we can then plug in our final value it says find the value of x when y equals eight so if we put eight in place of y we would have that x is equal to a half times 8 squared and we can work that out and we get the answer 32 6 8 squared is 64 and half of that is 32. again you can test this out on a calculator if you're unsure but again if you are needing to do a bit more work on direct proportion again the full video is in the description now we don't just have direct proportion we also have inverse proportion and the formula changes ever so slightly as instead we have x is going to be equal to k over y to the power of n again this is where x is inversely proportional to y to the power of n but the process is very similar and again that is all linked in the description on to our next formula we're going to have a look at when we're looking at a histogram and we're looking for frequency density now when we are looking for frequency density we use the formula that frequency over class width is equal to frequency density and when we are looking at this you would normally be given some grouped data in a table like the one shown on the screen now the frequency is shown to you very clearly in the table and the class width you normally have to work out and the class width is the distance between the two numbers there in the group data this is a particular example the frequency is 8 and the distance from 0 to 10 is 10. so we would do 8 divided by 10 for our frequency density for that particular column that particular row sorry and we would get the answer 0.8 now you would have to go and work out the frequency density for the rest of those rows but that would be how you would work out the first one that would allow you to then plot the height of your bars for your histogram and move forward with the rest of the question but also not forgetting that this formula can be rearranged if you are working backwards with a histogram and again if you're not sure on any of that i will link the full video in the description for when you are wanting to look at histograms now we're going to move on to our final formula for this video and that is looking at some coordinate geometry so we're going to be specifically looking at the equation of a straight line now the equation of a straight line is written in the form y equals mx plus c we also need to know a few other formulas for this and that is looking at the gradient of a line and the gradient of a line can be written as m is equal to y2 minus y1 over x2 minus x1 there are some different ways of writing that you can write the change in the y-coordinate over the change in x-coordinate or you could use the words rise over run m is the letter that we use for gradient and also you could potentially have to look at in the harder questions perpendicular gradients and you need to know that perpendicular gradients are the negative reciprocal of the other gradient shown above so once we've worked out a gradient if we're looking at a perpendicular line we do negative one over that gradient so let's look at an example question that involves all three but again all of these topics will be linked in the description for you so we'll say that a line passes through the coordinates a which is 1 5 and b which is 4 11. and we're going to find the equation of the line perpendicular to this line a b that passes through the 0.67 so the first thing we have to do is work out the gradient and if you look at the points a and b you can see that the y-coordinates are 11 and 5. so if we did 11 take away 5 that would be our change in y and 4 take away 1 would be our change in x you can do that in either order as long as you follow the same order for both of those numbers so our gradient would be 11 over take away 5 over four minus one and that gives us a gradient of two now we're going to be looking at the perpendicular line so the perpendicular gradient would be negative one over two or minus a half and you can see that i've put a little p there with the m which just explains that that is the perpendicular gradient onto the next part of this we'd look at the equation of the line so now we can put our gradient in we can say that y is going to be equal to minus a half x plus c we now want to substitute in a coordinate so you have to pick the coordinate that the line passes through and in this question it says it passes through six seven so if we sub those numbers in we get seven the y coordinate is equal to minus a half times six which is the x coordinate plus c we can then simplify the negative a half times six which becomes negative three and you can add the three over to the other side to isolate c and you get a value of ten now you have the value for c which is the y-intercept you can put that back into the formula shown at the top or the equation shown at the top i should say and that would give you a final equation of the line which is y equals a half x plus 10 and that would be everything you would need for looking at coordinate geometry and the equation of a line so that is just some bonus formulas there there are other formulas that you could have a look at for particular topics but these are the key ones you need to remember for the higher tier paper so i hope you enjoyed that video hopefully it was useful and it was helpful as always please do leave a comment please do like the video and don't forget to subscribe to the channel and share this with your friends don't forget to go into the description and check out any lessons on any topics that you are unsure of but there we go and i will see you for the next video [Music] oh [Music]