 Hello friends welcome again to another session on trigonometry and in this session. We are going to study one more Formula, which is called cosine formula in the last session You saw sign rule where a by sine a is equal to b by sine b is equal to c by sine c And that was equal to twice the circum radius of the triangle, right now in this session We are going to study one more Formula which is a square is equal to b square plus c square minus 2bc cos a and likewise for other sides as well So there are these these formulae are part of nothing something called cosine formula. Okay. Now, how do we even? Prove it. So how do we prove it? so there could be two cases guys one is when the triangle is acute angle triangle and The other one when the triangle is obtuse angle triangle So we'll try to prove this for both the cases. So let's try and first prove for case a that is it is acute angle triangle Right acute angled triangle. So if the triangle is acute angled then what happens? So clearly you can see the diagram here a bc is a triangle a small a small b small c Depicts the sides opposite to angle a angle b and angle c respectively So I have dropped a perpendicular from B on to AC. So here BD is so BD is perpendicular to AC here Similarly in this case also BD is perpendicular to AC right both the cases. So let's take up the acute angle case first Now first is a square is equal to b square plus c square minus 2bc cos a now Okay, so considering triangle B DC, okay in triangle B DC We can write by Pythagoras theorem a square is equal to BD square plus DC DC square, isn't it? right BD square plus DC square by what by Pythagoras theorem very easy no problems here by Pythagoras theorem I can write this isn't it? So hence a square is BD square plus DC square therefore a square is Equal to now. Can I not write a square as or sorry BD square as C square and it will be a better idea to write it as let's say this is x so this Let's say ad is equal to x Okay, so DC will be equal to clearly ac Minus x that is B minus x Okay, so let's write a a square is equal to a BD square plus DC square right, so let's first try and Write a few more equations relations. So like c square will be equal to BD square Plus x square by same logic by Pythagoras theorem Pythagoras theorem, isn't it? Right c square is equal to these BD square plus x square, right and ad upon ad upon ab is equal to cos a Is it ad? Upon a b is cos a because you know this this angle is a so hence BD upon a b will be cos a This implies ad will be simply a b cos a a a b a b cos a Now ad was my friend ad. What do we what did we assume it to be x and a b is anyway c so c cos a x is c cos a right x is C cos a now continuing here. What do we get? So clearly? BD square is equal to c square minus x square from here BD square is c square minus x square isn't it now c square minus x square and x is c square cos square a Right here is x Isn't it? And what about So BD square is no now DC square. What is DC? DC square is clearly b minus x whole square Right B if you see DC square is B minus x whole square. So if you expand it, it will be b square plus x square minus 2 b x Correct, so let's write this so b square and x square is c square cos square a from here again And this will be minus 2 b and x is c cos a C cos a so let me write the numbers of the equation. This is equation number one This will be equation number two and this is equation number three So from one two and three guys from one two and three What do we get? We get a square is Equal to let's write BD square first. So BD square is c square minus c square cos square a Correct and let's write DC now DC square DC square is given as B square plus c square cos square a Minus 2 BC cos a Correct from equation number three. So if you see this cancels out. So hence we get the first relationship a square is equal to B square plus c square minus 2 BC cos of a Right, this is cosine formula Now you can similar by similar logic you can prove the other ones too, right? Since the choice of a and b and c is totally dependent on the user or the person who's doing it So hence irrespective of whatever a b and c you can Apply the symmetry logic and you will get that These relations the second one and third one will also hold you can try on your own and you'll see that by the same logic instead of Dropping a perpendicular from B. You have to now drop from a and then from c like that and then you can find out You know the other two relationship as well. So this will pass through the same point Yeah, so this is the cosine formula for an acute angle triangles. Now. Let us see Will it be same for the obtuse angle case? So here is the obtuse angle and now let us again do the similar exercise and What you can see BD has been dropped again BD is perpendicular to AC. So hence here I Can I should write BD perpendicular to AC? Okay, so now What is AD so if you now say see again, we can write from here also a square is equal to BD square plus CD square by Pythagoras theorem, right? BD square can be written as A BD square can be written as C square minus AD square BD square can be written as C square minus AD square and CD square can be written as AD minus AD minus B whole square Am I right AD? Minus B whole square, right? Now if you see this is C square minus AD square then plus AD square Then plus B square minus to AD into B, right? So AD square AD square goes it becomes C square plus B square minus to AD into B nobody's AD guys so if you see AD upon C is cos of A, right? So AD will be clearly C times cos A So you can deploy this back into this equation to get the desired result So you will get a square is equal to C square plus B square minus 2 Times C cos A times B. So rearranging you will get a square is equal to B square plus C square minus 2 BC cos of A correct if you see guys, this is you know Pythagoras theorem becomes a special case for all of this What happens if A is? angle A is 90 degree so if Angle A is 90 degrees And what happens we know that cos of 90 degree is equal to zero, right? So clearly if A is 90 degrees, so what will the diagram diagram will be something like that, isn't it? So A becomes 90. This is A and This is B and this is C. This is Pythagoras theorem So that means this is A. This is B. This is C So a square must be equal to B square plus C square, which is true If you deploy this back here, you will get a square is equal to B square plus C square, right? This is the Pythagoras theorem. So you can treat Pythagoras theorem as a special case of cosine formula. So please remember these three formula for problem solving in olympiads and in different exams like J and all this is a very important formula This will be used in physics as well a lot. So in vector algebra, these kind of relations are used a lot. So please Keep these three in mind, right? This is called cosine formula. In the next session, we'll see projection formula