 Hello friends so welcome to another session on problem solving in this session We are going to solve a heights and distance problem in trigonometry so we had discussed earlier that heights and distances are nothing but application of trigonometry and it is widely used to measure heights of Towers trees poles and you know buildings without actually, you know climbing that particular structure and Measuring the height. So here is another question on That topic so it says a man wishes to find the height of a church spire So if you see he is willing to find out the height of the church spire It stands on a horizontal plane now you may ask why is that needed regulation and it is said that a Particular building should not be beyond a given height Then how to find out the height without actually climbing on that onto that structure So you can deploy this methodology to find height of any building. So what he does is he He sees that on a horizontal plane at a point on This plane he finds the angle of elevation of the top of the spire is 45 degrees So if you see I can have mentioned that as alpha is equal to 45 degree here So that is the angle of elevation of the church Spire, you know, so let us say RQ here in that in this figure is denoted by Let us say the church spire and PQ is the Lane on which the guy is and at point P He is seeing point R to be at an angle of 45 degrees from the ground now He walks 100 feet towards the The church and towards the church mind you So as you go towards the church the angle of elevation will increase and hence it is it is observed that now at point S Which is around 100 feet away from P. So this this distance is 100 feet away from P And now the angle of elevation has become 60 degrees. Okay Now you have to find out the height height of the tower and also his original distance from the foot of the Spire right so these two things have to be found out. So now if you see I have mentioned PS is 100 feet Right, so that's what has been given now clearly RQ is perpendicular to PQ RQ is the height and you have to find that out and it's given that to angle of elevations R P Q is 45 degrees and RS Q is 60 degrees, right? Now what will we do? We will now write and use our trigonometric ratios knowledge and write to and will try to write a few equations so if you can see in triangle RQ s that is a Right angle triangle at Q tan beta is given by H upon X, right? And that is opposite by adjacent and hence after rearranging you can write X is equal to H upon tan beta And this is this is let us say equation number one Now come to the second Triangle that is RQ P here also if you see tan alpha, which was given as 45 degrees is H upon X plus Y H upon X plus Y opposite by adjacent, isn't it? That means X plus Y can be written as H upon tan alpha and this is equation number 2 you can rearrange to write X plus Y as H upon tan alpha Now why are we doing it? So basically we want H, isn't it? So our objective is to find H always keep that in mind And X is unknown, right? So I will try to eliminate X Y is known because Y is given as 100 feet, right? So we have to eliminate X. So hence I'm trying to do something so that X can be eliminated So what do I do? I do this operation equation 2 minus equation 1 and hence if you see X gets eliminated because of that subtraction So hence I am left with all knowns and only one unknown that is H So hence if you do the simplification You will get H is equal to Y upon 1 minus tan alpha minus 1 by tan beta And now when you deploy values, it's always good practice to deploy values towards the end if the problem is not that complicated Why because it will try it will you know help you eliminating Error silly errors which you do while calculating, right? Also another thing to be noted here is please remember the basic standard Angle ratios so for example tan 45 has to be 1 you must remember this otherwise you will not be able to solve the problem Similarly tan 60 is root 3 and this particular knowledge After application you will get what? You will get this This calculation 100 times root 3 divided by root 3 minus 1 and if you see this value is nothing but 200 and Uh 36.6 feet so hence answer is Answer is 236.6 feet So this was part a now there was another another part to the question the part another part is and also is original distance from the foot of the spire So basically you have to find out X plus Y correct X plus Y Or if you see in triangle rpq So let me just try it here. So another part of the the question was What is the distance X plus Y? So if you notice X plus Y is equal to H only why because triangle rqp is an isosceles triangle Isosceles triangle why because this angle is 90 degree here. This is 45 degrees. So this angle has to be 45 degrees So hence this side is equal to this side That is pq is equal to rq hence Hence pq, which is the second part of the question is nothing but H is equal to 236.6 feet Right. So both the answers are 236.6 feet