[Music] all right here we go let's take a look at the next section here we're going to talk about inverse trig functions which are pretty awesome uh we've actually done this before but we're going to go way more in depth so I want to kind of do a quick review to get us started here so I know you can do this first problem this old school geometry but let's just humor me for a second let's review this real quick so when I'm talking about a trig function I'm saying hey what trig function relates this angle to these two sides is it s cosine or tangent sure it's a sign we're talking about opposite over hypotenuse so we would set these bad boys up so we could find the missing side of the right triangle we were all happy everybody loved it but let's make sure we really know what that means what in the world is s of 42 well the S of 42 is just the ratio of the opposite side over the hypotenuse so it's just the that fraction so it could be any size triangle it could be this little baby right triangle with 42° it could be be a huge one there's a big 42 Dee angle with a right angle but it's that ratio of A over B or opposite over hypotenuse so any triangle they're all similar as long as they get that 42 degree angle they all make that same ratio and then a lot of times we can type oh I already did that sneak peak there but we can already type it in there uh this will give us that ratio it tells us the ratio is 66913 so uh that's what the sign of 42 is it's just telling you what that fraction is equal to so normally we kind of didn't really show that step we just when we were solving the equation we just kind of said hey I'm just going to Times by 9 times by 9 just to solve this bad boy and we kind of did one step but really we're saying hey the sign is like 6691 blah blah blah blah equals the ratio of the opposite side over the hypotenuse so anytime I want the decimal or the fraction I just put in the calculator took the sign of 42 and then we can solve and find that side awesome so what is the opposite of that well the opposite would be what if I don't know the angle but I know the two sides of the ratio so now it's still s but I don't know what sign is I'm saying hey it's a sign of some angle it makes 3 over8 so now I know that decimal you know 38 is just 375 so there's that ratio that we're talking about and we're saying hey what angle creates that so this case instead of putting the calculator we didn't just say the sign of we don't know the angle so any we don't know the angle we would do what we would do this inverse sign second sign so above it in blue it says inverse sign and we would type in that Ratio or in this case I'm just going to put that little decimal in there boom 22 degrees so what is the actual step here so what are you really doing you're actually taking this inverse sign of both sides you're going to say hey I'm going to take the inverse sign of this o can I slide this bad boy over here okay I slid that over just wi so I can sneak this in I'm actually took the inverse sign of both sides so you're taking the inverse sign of the ratio and then boom what happens here the inverse sign cancels out the sign just like think about if you have an X squar and you want to get rid of it what do you do you square root it they're inverses each other they cancel each other out that's exactly what we did here and I think I said that was somewhere around a 22 degree angle like that so that's what inverse sign is is when you have the decimal or the ratio you want to go back to the angle versus when I have the angle and I want to solve part of the missing ratio awesome great all right all right so let's go ahead and start to try to graph these inverse trig functions so to do that let's come over here and start with a regular trig function so s of X we're pretty good at I got a graph down here for you let's just talk real quick domain and range no problems with x's you can put everything in there we're going to say it's all real numbers I'm going to use the fancy R for all reels super cool this goes forever and ever and ever how about the range of this bad boy so it's limited right the lowest it ever gets is negative one the highest it ever gets is positive one so we say yeah we got this the y are all sandwiched between - 1 and one great so why do I have the graph because I want to get to the inverse graph so when I think about inverses what do I think I'm always thinking hey I'm just switching my x's and Y values they're inverses of each other so I'm going to switch X and Y so when I do that graphically what's going to happen well maybe you already noticed check out my x axis oh it's the y- axis now and check out my y- Axis the x-axis they switch just like your range becomes your domain and your domain becomes your range they switch X and Y inputs and outputs so what's that going to look like well imagine this graph we can do a couple points here I'll do 0 0 I love that point switch Z 0 you're still 0 0 how about this point this point is over pi over two up one I'm going to switch that so I'm going to go what over one up pi over two so see how the value switch this is over one up pi over two and then we can kind of plug and chug the rest of the points here as I get it over here it's going to come over here get down here to netive - 1 negative 1 is going to make three halves and then 2 pi is at zero or Zero's at 2 pi so you see this sign curve kind of coming through like oh my goodness okay so let's see the sign curve coming up down I guess it's left and right it's no longer up and down it's left and right and then the same thing if you want to keep the pattern going the negative direction I've got this going on forever curve like this so is that a function no way that's a function does not pass the vertical line test this doesn't pass the horizontal line test it's not invertible we can't invert this so I you could go ahead and say yes if we could do it we could get the domain and range here we would just flip-flop them but because that's not really a function we limit it to what we call the principal value so what we're really going to do is we're going to limit the area we're looking so let's go back to this graph we're actually going to take just part of this graph and I'm going to highlight it I love the highlighter well what part do we want well to make it past the horizontal line test or to make it invertible we are going to go from negative uh Pi / 2 to pi over 2 so we all just kind of agreed this is the there's a bunch of areas we can look at but this is the one we're going to take from netive pi over two to pi over 2 so what we're going to do is we're going to say hey I don't want all reals I'm going to say let's go from pi over 2 actually let me use x ah we're going to say from negative pi over 2 and X is sandwiched in between there to pi over 2 so we're going to limit where these come from so we're going to restrict the domain of the function so that we can invert it well what's that going to do over here that's now going to be the range and we call these the principal values so we all can agree which ones we're talking about we know what part of the inverted function we're going to be at and what is the domain the domain is still and this becomes oh not X anymore if I switched it what's this become the Y excellent now the range will still become the domain so I'm going to say the domain is still sandwich between -1 and one awesome so what's going to happen here let's graph this bad boy so if I'm only talking about this little piece right here that's all I'm talking about that's all that's going to show up over here so let's start at negative pi pi over 21 so that's going to flip to I'll write this down this is too much here so I went over piun over 2 this first point and then I went down one is that point right there so on my graph what's going to happen I'm going to alternate it I'm going to say hey1 makes over 2 and I'm going to go down to right here 0 0 super easy I love that one it looks like this and one makes rad makes pi over 2 just like one pi over 2 makes one over here so what's going to happen here I've kind of got this little squiggly curly guy right here so when we use the restricted domain this is actually what it's going to look like over here that's going to be our restricted values for that so when I do that it kind of relates back to unit circle we're going to we're going to do this more here in a second well what part of the unit circle is that we're talking about and I love my highlighter we're talking about from negative pi over two to pi over two so remember this in the unit circle this is 0 to Pi over2 this is 0 to negative pi over 2 it's these two quadrants so we're always when we're talking about the main values the principal values we're always with sign going to refer to quadrants one and quadrants four so going back to that unit circle I love the unit circle I hope you do too it's so fun so the restricted values are going to be this part of the unit circle because if we keep going around and around forever and ever we get this periodic motion goes forever and ever so we got to strict it restrict it down to this awesome what if you just want to double check it you can we can check the calculator out and uh it'll show us to make sure we're not crazy here uh just plug it in there so I want to say hey I'm really interested what is the inverse sign of X just be careful when we're doing this we're always going to be in radians so check your mode I happen to be in degrees make sure you're in radians change that calculator over and then anytime you're doing trig just zoom seven Zoom trig it gives you a nice window and you can see it's just that little Squiggy line now really that would like to go on and up and down forever and ever just like our picture over here but we're restricting it to that dude that's awesome let's do two more of these bad boys what about cosine so we're gonna again we're going to restrict the cosine I I could graph this forever and ever and ever but I don't want to do it let's let's start with that and then I'm going to erase it so you may want to do this lightly so I can see this starts at 01 so I'm going to start at one Z so it's a good practice to invert these points this is pi over 2 0 so I'm going to go 0 pi over 2 and then this is what pi and -1 so I'm going to go -1 and pi and then once you see that you can probably see the pattern it's just like you're finding your points in the opposite direction you're going up and down instead of left and right and then we've got this nice curvy curve going on here there it is we got take a break all right keep it going again awesome but not a function so what happens here well it'll make it easy for you we got to go ahead and and restrict the domain again but this time we're going to say we're going to use this chunk right here so we're going to go from 0 to Pi that's going to be our restricted domain so it kind of changes things here so we are actually going from zero to Pi excellent and so I got to trim up my graph here so I want to erase uh what this is the domain so the range can only go zero to Pi so I'm going to stop here at Pi that's good I didn't like that part of the graph anyway get rid of that that was all squiggly well that looks pretty good and there's my little guy right there chilling he's good to go so we're going to restrict this domain if we're thinking about the old uh I thought I had a unit circle there it is the old unit circle what quadrants am I talking about here from zero to Pi is actually these first two quadrants so when I go to restrict the domain now and this is going to be important coming up here we are talking about quadrants one and two is my restricted domain and then that's what's going to show up on my inverse cosine check this out I put Arc cosine same thing so uh that is the exact same thing sorry if you're like what is he talking about uh has to do with the arc length of the unit circle actually so that's why that is there so we're going to say hey if I it's the same thing if I want to find the inverse cosine like this cos1 of X or you can say Arc cosine so we're going to use them interchangeably be be cool with both so same thing here this is just inverse tangent so don't freak out probably should should have led with that inverse tan is tan1 awesome or Arc tan fantastic let's restrict one more domain here so I graph tangent for us we could graph up all these they're basically going to be these same things sideways we could go forever and ever ever let's just go ahead and restrict it first so we know what we're talking about luckily this one is a repeat we pick the we pick this one right here on zero is the only one we're going to use because it passes the horizontal line test it allows us to go ahead and invert it which is fantastic so which one is that one again that is from where that's back to negative pi over 2 and 2 pi over 2 now notice it is undefined on those end points I get that um it is going to be that's the asop so it's kind of a nice thing for graphing so I've got this asint at pi over 2 and pi over two but instead of being vertical when I go to graph it what's going to happen it's horizontal remember I'm switching my X and Y so we've got got these asop at y y = pi over 2 and Y = piun over2 I can see 0 zos right here great point 0 0 now the question is does it start high and come low or does it start low and come high so let's just pick a point here if I go negative I can't get to pi over two but I'm way down here getting as I get close to pi over two so I'm I'm a bit this is getting more and more negative so let's switch it as I get more and more negative what am I approaching I'm approaching pi over negative pi over two so it does start low and then it goes High there it is right there that is the flipped values now I don't want to stack them not a function so I have the restricted domain how does that relate what quadrant is negative pi over 2 to pi over 2 let's get back to the old handy dandy unit circle and negative pi over 2 to pi over 2 is these two quadrants so jot this down have these we need to know these restricted domains we are again talking about one and four so s tangent they're the same we restrict them the same and then have a ballpark idea you have these graphs awesome let's see if we can put it all together including its graph this is the big dog right here I like it if this makes sense you're golden so um let's find the inverse function so how do we find inverse functions of anything well we switch our X and Y so the f ofx is the Y we're going to switch it with switch X and Y so here was the X's part of the sign I'm now going to make it Y and then I'm going to solve for y so we switch X and Y solve for y so can I solve this equation sure just subtract one from both sides we've got x -1 = 2 sin Y and we've got two s of Y right here I went ahead and put in parentheses you don't really have to if it's just by itself uh how do you solve this bad boy divide by two divide by two so we've got x - one all over two equals what equals the S of Y I left myself some room there strategically well played Mr brust because what am I going to do how do I get rid of a sign well I can unsign it I can inverse sign it so I'm going to inverse sign everything in here there we go and then whatever I do to one side I got to do to the other so you're going to get some weird looking answers don't freak out it's okay um they just are going to look a little crazy here so what happens here well inverse sign cancels the sign and I'm left with what's ever in there which in this case is just Y and then what happens here not nothing really exciting we're just going to leave it alone like that okay there it is right there so I'm going to go ahead and write it as with the notation because I kind of changed everything over to X and Y which is totally cool but uh let's go back to our our function notation here so I'm going to say the inverse function equals now this is kind of weird for me to see everything going on as is so I know it's the inverse but I've got this idea so I'm going to take this one/ half and put it out front is that I just want to make sure that's cool I don't want to lose you here instead of dividing by two I'm going to multiply by 1/2 so really in in the parentheses I'm going to multiply by 1/ 12 x minus one and there's a reason for that I picked kind of a hard one here for the notes we're going to we're going to bring the pain here let's just do it so why did I do that well now I can see everything going on so do we recall what a sign function looks like yes I know it's uh between NE pi over 2 and it's crosses at zero I remember this and I'm going to hook you up in the beginning as you get used to this that did not curve very well let's make that a little more curvy so it's the part of the sign curve if you have to Envision all going on between negative pi over2 and Pi over2 now let's talk about the parts we got some action going on here so what does this do well this is a horizontal dilation remember this is going to stretch it horizontally so I've got this horizontal dilation right here and remember it's the opposite of what you would think so it's going to be hor horizontal dilation of two boom so I got to take that into account what does this bad boy do over here ah it's a translation so I've got this horizontal it's going to shift it I'm just going to write shift if that's cool it's going to shift it right one but it's a horizontal translation of one so now what happens is uh you're gonna have to take your function and you're going to have to manipulate it so there's sign I'm sorry inverse sign I need to do a horizontal uh what do you call that stretchy stretchy so I'm going to stretch this out let's just change colors so you can see it happening so it's going to get twice as big so he was at netive 1 becomes -2 zero stays here this one comes out here like that so now I've got this stretch like so so I stretched it out after we stretch then we can move so we're following all our same rules from earlier in the year and now we're going to move it right one so we just move it over right one my dots did not want to join it let's make sure they come with that's awkward without them and I get this maybe I need to come right a little bit B more there it is right there holy cow so I stretched it and I shifted it so in the beginning I'm going to hook you up with the parent function and let you kind of manipulate it but it's all the same rules it's just that one little line and you got to move it around dude that's awesome all right now that we have it graphed let's take a look at the domain and range of the inverse function so this is pretty cool we know the domain of the original function so what's going to happen for the inverse function well it's going to become the range of the inverse function so that one's pretty chill man we just know this domain actually becomes the range of the inverse functions that one's that one's pretty chill how about this though we got to find the domain of the inverse function that's our goal we're finding this inverse function here domain range well we already found it we already graphed it but I just want to make sure we can list the domain arrange so I know we could look at that graph and figure it out but every time we're not going to graph it so let's do it algebraically here so how am I going to get that domain no problem we're just going to plug these in end points in so we know that it's invertible on this that's why I pick this um restricted domain so all I got to do is plug the end points in that's going to be my Max and my Min so first let's plug in uh the left one so we're going to say -2 sin x but in this case we're going to say x is pi/ 2 boom plus that one and then we'll do the same thing we'll plug pi over 2 in and find it let's go ahead and write that out then we're going to come over here and find the other end point we're going to say 2 s of positive Pi / 2 and that's going to say plus one so that'll give us our Max and our Min here so what is the S of pi/ 2 remember that's just our 90 that's a rough rough X and Y axis there but where am I at I'm down here at what 01 so or 0 negative one so I know that uh the S of piun over 2 is 1 add that one to it I'm going to say -2 + 1 is1 awesome can we do the same thing over here I'm going to say two s the two I'm sorry time the S of Pi two that's this point up here which is going to be positive 1 and then I'm going to add the one to it so 2 * 1 + 1 is three so that is going to be my new what domain so it was the range of this function you found the range of the function so then it'll become the domain of the inverse function so from -1 x is stuck between1 and three so is that cool so we just graphed it let's pull that picture back up that we just graphed we just graphed it is that our true domain and range did we do it correctly uh hopefully so yeah what's the domain from -1 to positive3 so there's the domain and where's the range from negative pi over 2 to pi over 2 so we're going to do it algebraically just plug and chug get those numbers awesome last part of this let's change gear so that's what inverse functions are why do we need them well they're going to we're going to use them a lot the rest of the chapter and we're going to solve some equations coming up so we have to take a look at uh what's going on here so if I want to find the inverse sign of 1/2 I like think about what am I doing I'm thinking about hey what angle can I take the sign of that will give me a ratio of 1/2 so what I like to do is go to my handy dandy unit circle there it is right there and what can I do with this I zoomed it in we kind of have to have this memorized we just got to know these different parts so don't freak out if you need to use the beginning that's okay but then you really got to start to work to the point where you just know what these are so do you know what uh sign value makes 1/2 what angle makes 1/2 well it I can see it right here remember it goes XY is cosine s and so I'm looking for there it is right there there's the sign and that is pi over 6 is that the only place where it happens no it actually happens over here doesn't it if this is pi if this angle's pi over 6 this angle is also pi over 6 less than Pi so it's 5 pi over 6 so it happens again it actually happen happens twice but we got the restricted domain between netive uh Pi / 2 which is down here and uh positive Pi over2 so when we're talking about inverse sign there's that highlighter woo we're talking about these two sections right here that's the only place I can pull them out so when you're you do your calculator it's going to tell you only from its restricted area here so if I have the restricted area it's quadrants 1 and four so now I know I'm talking about pi over 6 not 5 pi over 6 so that is called the answer here let's just write down so we have it is pi over 6 that is called the principal value the main value the one that comes from the invertible function uh so we know which one because technically is infinite we can go on and on and on and on about that forever and ever and ever awesome okay let's try number two so we're thinking about inverse cosine uh remember it's X comma y cosine comma s so where do I get a negative it's got to be the negative so I see radical 2 over two right there but I need it to be negative this is where I'm going to get the negative because the cosin's first so that's PI over4 what's his friend over here well this is going to be what 3 Pi over4 so if you have a hard time thinking about this just think about what would make 1 Pi 4 pi over 4 would make 1 Pi I'm pi over 4 less than that so I'm one less I'm 3 Pi over4 so there's where it's going to happen so it's got to be 3 Pi over4 because it has to be the negative that's the key and when I'm talking about cosine I've got a restricted domain where is it where's my restricted domain it is in quadrants 1 and two so when I think back to that graph remember I was only graphing in these two quadrants so the restricted uh value the principal value that comes out of that it's got to be up there it can't be the positive one it's got to be the negative one woo that's intense lot of unit circle hope you like the unit circle ah what about tangent tangent is tricky I'm not going to lie so if you don't have that the Circle there you can draw your own unit circle don't freak out you can make your own just a just a rough sketch is cool now tangent is what tangent is really uh s over cosine so we're looking for hey the tangent is s over cosine I need that to turn into radical 3 uh over one so what possible combo does that well uh that's going to happen when I'm looking for 12 radical 3/ 2 now why do I know that well I just stack them so radical 3 over2 if I stack them if I say s over cosine I know radical 3 over2 over2 has the same denominator there right so they're both two so they cancel and I'm left with radical 3 so in this case when I'm looking for radical 3 I'm looking for the S value to be radical 3 over2 the cosine over that so where does this happen where do we get that that should be up here at what that's pi over three so at pi over three and then again we'd have to be careful because we could have the negative version of that it actually happens twice here in here but remember this is the plus minus so if you had a negative denominator you'd have a negative radical 3 that would be a bummer not so much a bummer as that would be a different problem a different answer so I have to look in the restricted area again tangent is going to look in one and four in this area I have to find the right sign of it it's the positive version so that's I'm up here in the first quadrant not the fourth quadrant so there's choices don't pick the wrong one you have to think about it take your time to come up with these answers that was intense we need practice on that that'll get easier and easier the more and more and more you do awesome all right that's it we just knocked out inverse trig functions that was pretty awesome uh good luck on the practice and the master check peace out