Hi students in this module from the chapter Heights and Distances Let’s  show a sum based on Application Of trigonometry It’s going to be very interesting Let’s see the sum first which is Two vertical poles are on either side of a road A 30 metre long ladder Is placed between the two poles When the ladder Rest  against 1 pole It’s makes angle 32 degree and 24 minutes with the pole  and When it is turned to rest Against another pole It makes an angle 32 degree and 24 minutes with the road Calculate the width off the road Now this is what the sum says Right Let’s understand what are we supposed to draw It says the two vertical poles are on either side of the road You have two poles on either side of the road A 30 metre long ladder Is placed between the two poles You have a 30 metre long ladder Length of the ladder is given to us as 30 metres Now this is placed in such a way that  when ladder rests to 1 pole It makes angle of 32 degree and 24 minutes With the pole Right when it is placed On one pole The angle made by the ladder with the pole The angle there Its nothing but It is 32 degree and 24 minutes And when it is turned To rest against the another pole Now its rest on the other pole Here here It makes 32 degree and 24 minutes  with the road Right very important Thing to be noted there When it is rested like this The angle made by the ladder with the road Its nothing but 32 degree and 24 minutes Very important thing right first with the road then with the pole Right The sum says find Calculate the width of the road So we need to find the width of the road And we say the width of the road We are saying find the length of BD That is what we need to find So this is what we have in the sum Let’s think how I going to get this It’s going to be very easy Very simple Let’s see the sum Right here we going to describe First will say that AB and DE represents the two poles That is what we have there Right Then we have AC and CE It represents the two positions of the ladder Right AC Is the first position and CE is the next position they represents the two positions of the ladder Right and we have AC=CE=30m This is known to us And we have those two angles there angle BAC is 32 degrees and 24 minutes and angle ECD has 32 degrees and 24 minutes That is what we have we need to find BD observe BD look at BD BD is the width of the road right BD is nothing but made up of two things BC and CD so we need to first find the value BC then the value of CD lets find out BC you know the BC belongs to the right angle triangle ABC consider that right angle triangle ABC infact we have an acute angle yes let’s look at that for that acute angle BC something which we need to find that appears in the opposite side and what given to us is AC is the hypotenuse we are taking about opposite side of hypotenuse right opposite side and hypotenuse which ratio comes to your mind which trigonometry ratio yes its nothing but sin so were going to use sin for that angle so you’re going to use sin of angle BAC = BC/AC so you know the sin of BAC means it is sin of 32 degree and 24 mins that’s equal to BC if you don’t know keep it as it is AC we know that is 30 so BC upon 30 right beautiful now we are going to get that value of sin 32 degree and 24 mins from the trigonometric table from natural sines from that table if you see in that row which contains 32 and in the column which headed by 24 mins see there the intersect of the two will give you the value of sin 32 degree and 24 mins which nothing but 0.5358 so we get that value substitute the value there so we got the value of sin 32 degree and 24 mins that is 05358 substitute that now the value of BC you don’t know keep it as it is and now we know that it BC/30 now BC = 0.5358 * 30 so now we just multiply it and the product which we get is the value of BC that is nothing but 16.074m you got the value of BC now we need to find the value of CD observe CD CD belongs to the right angle triangle CDE right we going consider it right triangle CDE in that right angle triangle CDE we have an acute angle that is 32 degree and 24 mins for that acute angle we need to find the adjacent side and what is known to us is the hypotenuse so we are taking about adjacent side and hypotenuse when talking about adjacent side and hypotenuse which trigonometric ratio comes to your mind yes it is nothing but the trigonometric ratio is cos so your going to use cos for that particular angle so we right cos of angle ECD = adjacent side that is CD upon the hypotenuse that is CE now we just need to substitute the value so we write cos of 32 degree and 24 mins = CD/CE which is nothing but 30 so now we need the value of cos 32 degree and 24 mins this is something which we can get from the table of natural cosines so now look there in that row consisting of 32 and that column headed by 24 mins observe the intersection of that two gives you the value of 32 degree and 24 mins which 0.8443 so you substitute the value there 0.8443 = CD/30 now it’s so simple now we get CD = 0.8443 * 30 now we get CD = 25.329m so you got the value of CD too wasn’t that easy now isn’t it easy to get the value of BD yes it is so simple now we can say that BD = BC + CD the value of BC if you see its simple it is 16.074 + the value of CD we know it is 25.329 add those two and we get the value of BD as 41.403 and if you round it off now we can say that BD = 41.4m that means width of the road is 41.4m was that easy a very important sum but a very simple one.