Intro to Pythagorean trigonometric identities

now if you want to get introduced to trigonometric identities one of the best places to begin is with a right angle triangle and the moment you see this right angle triangle an equation a theorem pops into your mind doesn't it a square plus B squared equals C squared even if you try to forget this equation you'll never be able to it's that deeply ingrained in in all of us so much so we love this triangle so much that we even named the proportions like you saw on some of the previous videos if this angle is Theta we named this side the opposite side by the hypotenuse as sine theta the adjacent by hypotenuse cos theta and the opposite by adjacent side the lengths of those Dan theta now I want you to pause right now and notice that there is a difference between this equation and these equations and what is that it is that this actually says something about the world it says that the square of this site was the square of this side gives the square of this side but these equations are just definitions we have not told anything new about the world by just saying hey I'm gonna call B by C sine theta and if somebody tells you that you're like okay sure call it whatever you want to but the next question you need to ask is why is that useful anytime we name anything it's almost always because we find this this way of thinking about it to be useful and the question should be how exactly is it useful now you spend some time familiarizing yourself with what these ratios are and we're not going to slowly enter the domain the area where they start becoming useful now to do that the first challenge to you is can you state this like this is a statement right like this length squared plus this length square is this length squared written in the language of lengths lengths of the sides can you convert this into the language of trigonometry can you state the same sentence can you say the same sentence using sine theta cos theta and tan theta and if you have to how would you do that think about that what you see here is a square B Square C square but what you see here is B by C a by C one way in which you can actually show or write the sentence in this language is to somehow make this have B by C a by Z etc because then you can write sine cos and all that over here and how can we do that now one way to do that is okay so there is no a by C here but the beauty of math is I can make if I see when I don't have it and the way I'll do it is by dividing both sides of the equation by C square now let's see what happens if I do that if I take a square plus B Square and then I divide both by C square I can just take it separately to both of them by C square and that will be equal on this side to C square by C square now notice that a square by C square is simply a by C the whole square which is cos theta square cos theta squared now notice that we don't write cos theta squared but that's a different thing that's cause of the square of this angle some other triangle has theta square as the angle that'll be that so we put cos square theta what we mean is is this ratio a by C squared so it's cos square theta plus B by C is over here and that's sine theta so you get sine square theta and that's equal to C square by C square and that's equal to 1 now this may be the most famous there is a celebrity among trigonometric identities now used a new word this is a trigonometric identity what do I mean by that what I mean is that this equation that you see over here it's true for all values of theta it will not matter whether you put 30 and there's a 45 in this and currently in our world all the theta can be is vary between 0 and 90 so whatever value you put this will work but also notice that this is nothing different from a square plus B squared equals C squared it says the same thing it just says Pythagoras theorem this is Pythagoras theorem written in the language of trigonometry now now that you've done this maybe a question that you have is okay I wrote it in the language of trigonometry but where is tan theta isn't tan theta are gonna feels bad does it not have an identity for itself well let's let's give it one and the way you can do that is if I were to bring this over here the way you can do that is ask how do I bring B by a into this equation you can just divide this B by a and you will have it and let's see how do we do that so let's say I create a square plus B Square and this time I divide both sides by a square instead of C square over here so I'll have a square by a square plus B square by a square that will be equal to C square by a square by a square now once again you can look here and see what these all will mean so a square by a squared will simply be equal to one and then B square by a square and you see over here B divided by a so that this happens so this is B tan of theta squared or tan square theta 1 plus tan square theta and that's gonna be equal to C square by a square now you look up here and you don't see C square by a square right you only see a by C which is cos theta but you know that C by a is the reciprocal of cos theta which is what you have over here but what is the reciprocal of cos theta that's secant theta so you can either write 1 by cos theta square over here or you can just directly write secant theta I'm just gonna gently move this over yourself space to write secant it out here so secant theta square and I can maybe put a box around this as well notice again that this is not saying anything different or anything more about the world then Pythagoras theorem already did it's just restating it using new words now let's look at what else we can do we divided by C square B divided by a square why won't we divide by B Square just in the interest of seeing what what really happens let's do that so you have a square plus B squared equals C squared you can take all three and divide both sides by B Square and I would like you to just stop and do this right now and see what you get I'm gonna do it now so a square plus B squared equals C squared and I'm gonna divide both sides by B squared this time not a squared but B squared so B Square B squared B squared now notice again that this becomes 1 but what is a square by B squared let me look over here I don't see a square by B squared but I see it's reciprocal B by a which is tan theta which means this is going to be cot theta right so I can just write this as cot square theta I could have written 1 by tan square theta but I think cot square theta is a shorter way of writing it plus 1 equals C square by B Square and once again I don't see C by B but I see B by C so I get sine square theta sorry cosecant square theta or 1 by sine square theta so I have this as well so all these three are pretty much the same thing if you notice you could you can just move it across between them they sing the same thing in the language of ratios a small detail here is that when you look at cosecant square there's actually 1 by sine square cot is caused by sine for these to be valid the denominators should not be 0 and you may see that if 1 by sine Thetas over here and if sine theta is like 0 which is possible when theta is 0 then this equation won't work so this equation or this identity works for all the values of theta except when theta equals 0 and if you come over here tan theta is just signed by cos right and secant is 1 by cos so if cos theta becomes 0 this equation or identity will become invalid and when does cos theta become equal to 0 it becomes 0 when theta becomes 90 degrees right so for 90 degrees alone this won't work so just because they don't work for a single value of theta do we want to throw them away we don't want to write because they work for every other value so we like to keep them as identities but if you notice this one here this works both for 0 and 90 because there is no concept of denominator here that's just a small detail though because when you learn more about trigonometry in the full version the picture will expand from just ratios of sides of a triangle and you'll notice that theta can not just be between 0 and 90 it can be I don't know once three or something and you will have new ways to imagine what the sign cos and tan are so right now we are in the division of trigonometry and that's good because now it keeps it simple and what you can do with these identities is move from one ratio to the other if for some reason you know only sine theta but you want to know what cos theta is then what you can do is use sine theta in this equation and then find cos square theta with it and then cos theta with it so that's the main purpose of these trigonometric identities to serve as building blocks for you to prove even more difficult looking statements these become the building blocks and as you solve more problems you know exactly what I mean to give you an analogy when you first learnt a plus B the whole square equals a square plus B square plus 2 a B that's a thing that you had to remember and you had to use to prove more complicated things somewhere in expression in an algebraic expression if you find a plus B the whole square you will write it this way or if you find this you may be able to move it to this now this is an algebraic identity and this serves as a building block for proving more complicated statements what we have here is the trigonometric version of these identities so that's how you call them trigonometric identities