hi everyone today we are going to look at bearings for CSEC mathematics it is a can an application of what we have lids in trigonometry such as sine rule cosine rule area of a triangle sometimes trig ratios even Pythagoras theorem but for today's examples and lessons we'll focus on applications of sine rule and cosine rule so cardinal points shows the four main directions on as shown in the diagram we have north south west and east in some questions we will be required to do a diagram in terms of if they tell you that a vehicle is traveling in the eastern direction or the southern direction you need to know in the sketch of a diagram which direction that is but for today's lesson the most important direction that you need to remember is north the North line is always pointing upwards what is a bearing so a bearing gives a direction in terms of an angle from one point to another or with reference to a particular point so the bearing of points B from point E that means what direction B is from point E however our direction in this case to be very specific we are not going to use north south west and east we are using degrees or bearings which gives us a more accurate direction so a bearing is a measure of the angle in a clockwise direction from the north line so if I'm measuring that angle at the point E this arrow the blue arrow represents the North line and if you're going to clockwise direction to get to the point B if this angle is 115 degrees then the bearing of the points B from point E would be 115 degrees so again a bearing is just the angle mid with the North line in a clockwise direct now if we are looking at the opposite if we are at B and we need to be able to get the bearing of the points e then the bearing in this case is a clockwise direction at the points B from the North line and a North line being drawn at the point because we are considering ourselves to be located at B if you want to get directions to get to a the North line here at our position we are taking in the clockwise angle from the North line at the point B so this bearing can be calculated it's not given in this particular example but it's very simple to calculate so calculating the bearing of a from B so this is a our previous diagram and just recall that alternate angles are equal alternate angles form a z shape and that means once we have a z shape in our diagram as highlighted with the red dotted line that gives us alternate angles so this line this angle 115 degrees would be the same as this angle at the blue arrow since alternate angles are equal so we are really calculating the size of this angle here represented by the orange okay so to figure out the size of this entire angle if you just look at this vertical line here on a straight line angles add up to 180 degrees so this shaded region in green would be 180 degrees and then this part of it is 115 degrees since alternate angles are equal so therefore the bearing would be the 180 degrees added city 115 degrees which will give us 295 degrees the bearing at the point B to the point a would be the size of this clockwise angle made with the North line this spot being 180 since angles on a straight line add up to 180 and this one is 115 since Altan angles are equal such as adding them gives us our answer I hope this is clear because we will need to apply this knowledge when doing bearings for other questions so in general a bearing is usually represented using three digits so if we have an angle that is just 35 degrees as a bearing you need to write it using three digits so we write to zero at the front zero three five means just 35 degrees but bearings must be represented or expressed using three digits in most cases questions involving bearings usually involve the sine rule and the cosine rule as I mentioned before and it can also involve other parts of tricks such as Pythagoras theorem oath or trig ratios but just to recap Pythagoras theorem and trig ratios are only used for right angle triangles they cannot be used for non right angle triangles okay example one we are going to look at a passed paper question John 2014 number 10 Part B the diagram below shows a position of three points P Q and R on a horizontal plane that just means a flat surface where P Q is equal to 120 kilometers as shown on the diagram here from Pizza Q then we have P are equal 150 kilometers and Q P R equal 23 degrees all of that information given is represented on the diagram the first part of the question wants us to calculate to one decimal place the distance Q R so as seen in the diagram this is the distance that we want to find Q R we have a triangle in which we are given two sides and an angle and we need to find a third side so just to recap from our previous lesson of sine rule and cosine rule when to use which one a cosine rule the cosine rule involves the length of three sides and an angle where as the sine will involve two sides and two angles so in this case can counting decide that we want to find as well our question involves three sides and the angle 23 degrees therefore for this part of the question we have to use the cosine rule so applying the cosine rule to any question means that whatever side we have written on the left-hand side of the formula is the same as this angle so this would be the side P and angle P and if we label in terms of the PQ 1 R as in the side Q R remember that this is a side opposite to give an angle alright so if this angle is 23 degrees this is the angle this is angle P this is um this is a side that represents that corresponds to that angle so this side on the left-hand side of the equation must be opposite of this angle that we are using on the right-hand side of the formula and these would be the other two sides that PR n PQ in any order you can have PQ and NPR it's just the other two sides so applying the cosine rule to this question we have PR squared from the diagram PR is 150 degrees 350 kilometers PQ is equal to 120 kilometers alright and this is just an arrow here in the typing of the questions is supposed to be squared 120 squared minus 2 by 150 by 120 cost of 23 degrees and working this out 150 squared gives 2 22,500 120 squared will give us 14400 simplifying inside the brackets 2 by 150 by 120 would give 36,000 and I'm leaving this as costs of 23 degrees I'm working on inside the brackets here to give this value 33,000 one hundred and thirty eight point one seven and I'm also adding these two values here to get the 36 thousand nine hundred and then last step would be to subtract and this will give us three thousand seven hundred and sixty one point eight three but this is not a final answer because you want the length of QR and on the left hand side I have Q R squared which means that Q R will just be the square root of this value which is sixty one point three kilometers to one decimal place and as a question specified at the top calculate to one decimal place that's how I knew how to round off if the question did not specify you can round off as you like but once it's specified you need to follow the instructions but B of this question would like us to calculate the area of the triangle PQR and we learned our formula for area of a triangle in the last video tutorial where we have areas equal to 1/2 a B sine C where a B are the two sides that are adjacent to the angle or touching the angles so if this is my angle 23 degrees the two sides that I'm gonna use would be the 120 and 150 so substituting into our formula we would have a half by 120 by 150 sine 23 which gives three thousand five hundred and sixteen point six kilometers squared to one decimal place but to find the bearing of P sorry the bearing of P from Q is given as 252 degrees calculate the bearing of our from P so looking at our diagram the bearing of our from P is shown by the red arc on the diagram we loop the bearing can be found by calculating the size of this angle and this is a red arc that we are referring to so the bearing of our from P means that if we are located at P we want to get to R we are using this angle clockwise angle with Northline so we need to find the size of this angle indicated by the red arc so in doing this we are using the information that is given in the question that the bearing of P from Q is 252 degrees so at the point q the bearing of P from Q means that we are at the point Q and this clockwise direction represented by the blue arc is that 252 degrees thus the bearing from off P from Q so given that this bearing is 252 degrees then if I use alternate angles represented by the blue dotted line this part of the angle here would be 250 to take away 180 which is equal to 72 degrees the reason why we are working this out is because to find the size of this angle here represented by the red arc we need to know that this part of it is unknown now this part of it is a 23 degrees as indicated in the diagram however we need to calculate the size of this part of it and if I use alternate angles then that's missing angle would be equal to this part of the angle so that's what we are working out in step 1 we are figuring out this parts of the angles so that we can know this one so again from the question is given that this entire blue arc here represents the bearing of P from Q which is 252 degrees and a straight line gives us 180 so this half of it is 180 so this little part would be the two 52 degrees minus 180 which is 72 so if here is 72 then since alternate angles are equal then this part of the arc the red arc would be 72 degrees as well therefore the total bearing would be the 72 degrees plus a23 degrees which gives 95 degrees and again a bearing needs to be expressed using three digits so I'm writing that as zero nine five degrees example two and not the past paper question June 2011 number 10 Part B the diagram below shows the roots of an aeroplane flying from Port City represented by the letter P the Queen stone represented by Q and anti Riversdale represented by are the bearing of Q from P is 132 degrees and the angle PQR is 56 degrees so all that information there is represented on a diagram we have the point P the point Q and the point R then the bearing of Q from P is 132 that is shown on the diagram as well that means if we are at P to get to Q our bearing would be 132 degrees and the angle PQR is 56 degrees also indicated in the diagram as well as they give us some additional information on that diagram the first part requires us to calculate the value of x as shown in the diagram and this is fairly straightforward because we know that on a straight line angles add up to 180 degrees so our solution would be 180 degrees minus the sum of the other two angles which is 56 plus 48 and we can put this directly into the calculator as shown brackets must be used for the 180 to subtract isang and that will give us 76 degrees so X would be 76 degrees part two the distance from P to Q is 220 kilometers the distance from Q to R is 360 kilometers calculate the distance R P so using the cosine rule well let's just identify why we need to use a cosine rule for this particular question so we are given two distances we have PQ it's not shown on the diagram but that disturb that information is given in a question so from P to Q is 220 and then from Q to R is 360 so we know two sides of the triangle and we want to calculate the length of the third side and we know the angle that's opposite that side that we want to find out therefore we are using the cosine rule since it's involves the lengths of three sides and one angle so using that cosine rule besides that opposite the angle 56 degrees would be this side RP and that's going on a left-hand side of the equation so we have our P squared equal PQ squared plus Q R squared minus 2 times P Q and Q are costs of that angle Q or in other words are P squared equal the sum of the other two sides squared minus 2 times the other two sides x costs of the angle so just substituting our values given from the question above we have from port city to queenstown is 220 so that means PQ is 220 so P Q squared will be 2 xx squared then from Q to R will be 360 kilometers so this would be Q R squared would be 360 squared minus 2 by 2 20 by 360 cost of 56 and just simplifying our values so sneeze face two teams will be simplified to give us one hundred and seventy eight thousand and what we have in the square brackets would be simplified to give eighty eight thousand five hundred and seventy six point one five six and then subtracting into the calculator will get this value eighty nine thousand four hundred and twenty three point eight four four and on this side we notice on the left hand side is our P squared not our P so to get the length of our P we need to find the square root of that value which gives us two 99.04 which can also be just rounded off to 299 kilometers there is question didn't specify how to round it off so we can round us off to 299 kilometers the term in the bearing of our from P so I've put in some of the information from what we had from Part two so in part so they had told us from Q to R would be 360 they also gave us a length from Pizza Q but that's not relevant for this part of the question and from our solution we have PR being 299 so I just added that into the diagram the bearing of our from P means that if you are located at P and you want to get to R we are taking the bearing which is a clockwise angle from the North line at P so that's represented by the Red Hawk here right so the calculate that bearing then the one that is two degrees from the diagram is not the entire red arc that just represents a bearing of P to Q so that's until this line here which means that we need to figure out this angle and inside of the triangle at P so to figure out this angle which is angle R P Q or just the inside of the triangle that part of the angle at P we are going to use a sign rule to figure out this angle and then once we know this angle then the bearing would be 132 degrees add this angle alright so how do we know that we have to use a sign rule well we have two sides and volts one two and then we have two angles involved 56 degrees and the one that we want to find however since we also know the length of the side PQ I believe the cosine rule could also do it for this one if we use all three sides of the triangle and we can also see cosine rule to figure out this angle at P right but for my solution I've chosen to use the same rule so using the sine rule remember that the corresponding sides are opposite to the angle so P over sine P would be 360 over sine of this angle I'm calling that P and then Q over sine Q so if this is angle Q I'm referring to the fifty-six degree you say we are looking at the triangle so we are referring to the angles within the triangle so the angle Q would be the 56 degrees the Q over sine cubed B 299 over sine of 56 and cross-multiplying will give us sine P by 299 equal 360 by sine 56 so making sign P the subject of the formula we would have to divide by 2 99 on the right hand side so that means sine P will give us 0.99 eat 1:7 notice I'm not running off too much at this point because if we run off to one decimal place or two decimal places here it could affect accuracy of our answer our final answer so we're not running off too much yet so making Peter subject of the formula we would have to take to invoice on the right hand side so sine inverse of that value gives us eighty six point five degrees the one decimal place in our final answer we can round off the one decimal place or two decimal places as required from the question but this question didn't specify so I'm just running off to one decimal place therefore the bearing of our from P as I explained in a previous diagram well previous page that the bearing would be this 132 degrees Plus this angle inside of the triangle at P and we have found that this angle on the inside at P is eighty six point five degrees so the bearing of our from P would be 132 plus 86 point five which gives us two hundred eighteen point five degrees so I hope you have understood the two examples that we have discussed if you have any questions feel free to contact me remember that bearings is an application of what you have learnt in tricks that could involve sine rule cosine rule which is the case most of the times and sometimes it can involve trig ratios and even Pythagoras as well so it's important to understand how to use the sine rule and cosine rule very well before you apply your knowledge to bearings and also to analyze a question carefully to figure out what exactly the question is is requiring you to do or where exactly is the location of the bearing or the angle that you are required to calculate so you need to analyze the wording of the question so for example the bearing of B from e means that you are at ease so from e to the point B the bearing would be the angle at E as you are from e alright so thank you for listening to my video tutorial and I will we will continue next lecture please message me if you have any questions thank you