 Welcome to this session on trigonometry and today we are going to take up heights and distances now heights and distances will be an area of application of trigonometry for example you can measure the height of a building measure the height of a you know long tower or a tree or measure distance let's say The distance between two banks of a river using the principles of trigonometry This was the same principle the same methods were used In calculating let us say radius of the earth or let's say a distance between two cities and things like that wider applications will be all of these now this height and distances The you will be seeing a lot of industrial applications as well. So hence it is very important to Understand the concepts of height and heights and distances So in this there are a few concepts, which will be discussing and then we will be taking up a problem and see how do we apply? The concepts of trigonometry to solve such problems. So So first concept is angle of elevation Angle of elevation Okay, so what is angle of elevation now? I have drawn a Representative diagram here. Let us say a be represents the ground. Okay, so you are on the ground and Let us say BC is a tree BC represents a tree Okay tree or tree or let's say at our Or a building a Building and whose height you want to measure Okay So let us say on the top the top point of the tree or the tar is point C and this angle This angle alpha Which is between the line joining the foot this point is the foot of the tree put on the ground on the same line on this line itself is it a foot of foot of the let's say the tar on the line joining fair where you are standing Okay, and when you're watching the highest point on C Highest point C the angle made is alpha now this angle is known as angle of elevation so let us say you your eyes at this this point and This is up This is up the line of sight is on the B point B Okay, now you start rotating this line Let us say a be so that so that it now points to see Right, so you basically took this line a from So sorry a be from this point or along a be line to along a C line So whatever the angle stripped by that line is called angle of elevation Okay, so now you understood angle of elevation similarly. There is another concept of Angle of depression Angle of depression. So what is angle of depression angle of depression is just nothing but from C point C now Let us say you are viewing point a Okay, and let's say CD represented the parallel to the ground line So how much? CD has to rotate to coincide with C a this line is called angle of depression So if you see initially The line of sight was like that and it was turn turn turn turn and then finally it is like that So this angle swept is called angle of Depression so with these concepts only you will be able to solve a lot of Questions, okay, so let us let us try and see one of the Problems how to solve heights and distances problems now here is one of the problems first example will be taking in heights and distances It says a vertical flagstaff flagstaff stands on a horizontal plane Okay, from a point it distance 150 feet from its foot From a point distance 150 feet from its foot the angle of elevation of its top is found to be 30 degrees you have to find the height of the flagstaff a typical height and distance problem So let us Try to solve this problem in any heights and distances problem I would recommend draw a Proper Representative diagram, so let us say this represents. Let us say this point is a and this point is B Let us say a B represents the height of height of the flagstaff Okay, and let us represent oh a and Way represents the horizontal ground now what do you do you just join OB? Okay, and it said that angle of elevation of its top so top is be obviously this point is the top and and The angle of elevation of this point is nothing but 30 degrees. So it's given alpha is equal to 30 degrees Okay, angle of elevation of the top from this point oh a and it is given that This is 150 feet away 150 feet away from the foot Okay, so this is the diagram now you have to find out H. I will simply apply the trigonometric ratios and You will be able to solve this problem. So we clearly know this angle is 90 degrees. Is it it? So hence we can say in triangle oh a B oh a B a B upon Oh a is Equal to 10 30 degrees Isn't it now a B is H and Oh a we know is 150 feet 150 feet and for and from trigonometric ratio tables we know and 30 degrees 1 by root 3 So in simply H will be equal to 150 into 1 by root 3 feet and this value is 150 into 1 point 1 by 1.73 feet, which is approximately 86.6 feet So this is how our trigonometric height in distance problem can be solved. So if you see What is the practical application if you see you don't need to you know go to the top of the building to measure its height you can you know you just measure distance from the foot of foot of the building and and you just Find out the angle of elevation of the top and with these two quantities you can find out the Height of the building, okay