Intro to the Pythagorean trig identity

so we've got a right triangle drawn over here where this base is length is a the height here is B and the length of the hypotenuse is C and we already know when we see something like this we know from the Pythagorean theorem the relationship between a B and C we know that a squared plus B squared is going to be equal to the hypotenuse squared is going to be equal to C squared what I want to do in this video is explore how we can relate trig functions to this to essentially the Pythagorean theorem and to do that let's pick one of these non right angles so let's pick this angle right over here as theta and let's just think about this what the sine of theta is and what the cosine of theta is and see if we can mess with them a little bit to somehow leverage the Pythagorean theorem so before we do that let's just write down sohcahtoa just so we remember the definitions of these trig functions so sine is opposite over hypotenuse cut cosine is adjacent over hypotenuse and Toa tan is opposite over adjacent we won't be using tan at least in this video so let's think about sine of theta so sine of theta I will do it I'll do it in this blue color so sine of theta is what it is opposite over hypotenuse so it is equal to the length of B or it is equal to B B is the length B over the length of the hypotenuse which is C now what is cosine of theta cosine of theta well the adjacent side the the side of this angle that is not the hypotenuse it has length a so it's the length of the adjacent side over the length of the hypotenuse now how could I relate these things well it seems like if I square sine of theta then I'm going to have sine squared theta is equal to B squared over C squared and cosine squared theta is going to be a squared over C squared seems like I might be able to add them to get something that's pretty close up of Aggron theorem here so let's try that out so sine squared theta is equal to B squared over C squared I just squared both sides so over C squared cosine squared theta cosine squared squared theta is equal to a squared over C squared over C squared so what's this sum what's sine squared theta plus cosine squared theta so sine squared theta plus cosine squared theta cosine squared theta is going to be equal to what sine squared theta is B squared over C squared B squared over C squared plus a squared plus a squared over C squared a squared over C squared which is going to be equal to well we have a common denominator of C squared and the numerator we have B squared plus a squared B squared plus a squared now what is B squared in it plus a squared well we have it right over here Pythagorean theorem tells us B squared plus a squared where a squared plus B squared is going to be equal to C squared so this numerator simplifies to C squared and the whole expression is C squared over C squared which is just equal to one so using the sohcahtoa definition in a future video we'll use the unit circle definition but you see just using the unit's just even using the sohcahtoa definition of our trig functions we see probably the most important of all the trig identities that the sine squared theta sine squared of an angle plus the cosine squared of that same angle I'm introducing orange unnecessarily plus the cosine squared of that same angle is going to be equal to one now you might probably say okay Sal that's kind of cool but what's the big deal about this why should I care about this well the big deal is now you give me the sine of an angle and I can solve this equation for the cosine of that angle or vice versa so this is actually a pretty a pretty powerful powerful thing and this is also part of the motivation even for the unit circle definition of trig functions