 Hello friends, so continuing with our trend on solutions of triangles in this session We are going to understand what is projection formula. This is going to be used a lot in vector algebra and calculus later on and Anyways in geometry and in trigonometry also This formula is very helpful in solving lots of problems. So let's understand what projection formula is and Why we why do we call it is a projection formula now projection formula talks about a triangle? And in triangle ABC, let's say a small a small b and small c are the three sides opposite angle a b and c respectively Then a is given as b cos c plus c cos b B is given as c cos a plus a cos c and c is equal to a cos b plus b cos a first of all understand how do we write it because That would be interesting, you know and to remember also you need some kind of a mechanism. So if I'm writing a Then while writing a there will not be any angle a right and it will be b times So a the other two sides are b and c so it will be b times cos c and c times cos b so you can see B and the other angle c right and here it is c and the other angle is b like that Now while I'm writing b there will not be any angle b on the right hand side So c cos a plus a cos c it's a cyclical stuff here Now similarly c is equal to a cos b. So if you're writing the side a then you have to take The other angle so there are two angles two sides in all, isn't it a b and capital a and capital b You have to just remember that cos is there in the Equation so a times cos b and b times cos a like that you have to remember Okay, and if you see it's very easy also to you know prove it and then hence No need to mug up mug it up as such you will solve multiple problems, and that's the best way actually to remember any formula Okay, so now if you see I can write a is equal to Bd plus dc, isn't it now to you know make it clear there are two cases This is case one where it is a it is an acute angle triangle a cute triangle and here it is an obtuse Triangle so in both cases the formula should work correct now I've dropped a perpendicular ad on BC in both the cases in this case It is BC extended in this case BC extended and ad is perpendicular to Ad is perpendicular to BC. Yeah, now if you see in case one a is equal to bd plus dc Now clearly if you see bd upon bd upon c is equal to cos of angle b, isn't it? So clearly bd will be equal to c times cos b correct and similarly in triangle adc if you see consider DC upon DC upon DC upon ac or b ac is b is equal to cos of c So this implies dc equals b cos c so from one and two directly you can say a is equal to bd which can be written as c cos b and dc which can be written as b cos c So hence this relation is proved right similarly the other two can also be proved very easily with similar arguments Let's see what happens in case of obtuse Angle now in this case. What is a guys a is nothing but dc minus dc Sorry dc minus db isn't it dc minus db right now dc dc upon dc upon Ac that is b is if you see it's nothing but cos of c right so dc can be written as b cos c b Cos c Am I right so dc is equal to b cos c right similarly if you now see db upon db upon C in this triangle. Let's say this angle is theta is cos theta now cos theta is cos of 180 degrees minus 180 degree minus 180 degree minus Angle b Is it so cos pi minus theta or 180 degree minus theta is minus cos b correct so hence db Db is equal to minus c cos b Is it my db is equal to c cos b so hence now I can find out a so a is equal to dc which is b cos c Minus db so minus db is minus c cos b Correct so hence you quit b cos c plus c cos b Is it so this is how if you see in all the two cases You know both the two cases actually the formula is same so a is equal to b cos c plus C cos b that's called projection formula. Why is this called a projection formula is simply this? that if you have imagined like that that if you have a floor and you have a Stuff or a you know linear object like that maybe a pen or a pencil or whatever and let's say you throw light from top Okay from top So what would be the shadow like the shadow would be something like that isn't it this much this much for This part of the so this will be the shadow length so if this is b and this angle is c Let's say angle c. So this will be b cos c is the shadow of This part which part let's say this is a so shadow of c a will be of length b cos c Similarly, you can extend it on the other side. Let's say this is b now This is b this angle is b and this side is c. Let's say so shadow of If you throw light here also the same, let's say that you have a laser light so the shadow length of a be will be C cos B So we say that this length c cos b is nothing but the shadow length is the projection of what a be onto this Line Okay, hence. This is called projection formula So hence projection of AC or in other words if you're a very you know if you're looking at a be AC from top It would appear to be of this much length this month length from top. So that's called the projection of AC Onto this line. Let's say this line is x x dash Okay, so the projection of projection of C a on x x dash is nothing but B cos c right so hence Because we are using the concept of projection here to express one side in terms of the other So hence we call it as a projection formula. Hope you understood this