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AP Calc Trig Review

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Unit Circle, Graphs of trigonometric functions, and inverses of trigonometric functions

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hello everybody today we're going to talk about something that's part of supplemental it is a completely precalculus topic this is not anything to do with calculus but I did want to go over it just in case or precalculus skills were kind of shaky I did see some need for review on these topics so I thought it would be good to create a video for you guys we're going to go over some trig Concepts and our objectives for this video are to remember how to construct a unit circle and really my goal here is just to remind you guys of where all of the values on the unit circle come from and how they're useful and remembering key components of trigonometric graphs I'll grab some things like sine cosine tangent and so on I also wanted to find period of a trig Mount part of me trig metric function and why it is derived the way that it is and then lastly I would like to Philip properties of the six inverse trigonometric functions and this is that objective if you remember from the last video that we saved for today alright so first things first we're going to construct a unit circle I've already drawn my circle cuz I wanted to make it look a little bit nicer I'd like to start off it's just adding and information that's really easy to ask this thing remember that a unit circle is defined as having a radius of one soon throw it on the coordinate plane and we sent right at 0-0 we know that this points  if we just use our basic knowledge of according plane has to be 1-0 this point has to be zero one over here we must -1 zero and lastly we must have 0 negative 1 nothing new there that's just the basics of a coordinate plane next week an ad in pretty easily are some radian values knows that for every 180 degrees right for every half of a circle we have one pie and I'm also going to write the degree value there as well under 80 degrees  are we know that for every 90 if we go around the circle here you would have 90° we would have halfway between here and here halfway between pi so we would hi  and if we go around the circle again another pie or we have to tie and then if we went half way if we went one and a half we would get one best pie or 3 pi over 2  and if we remember every 90° this is 270 degrees last Leaf we come full circle we get 360 Degrees okay not so bad  now let's start breaking this up into useful pieces and the values were going to break it up into are going to lead into ice review special right triangles answer the degree values were going to use and every little quadrant here are going to be every 30 every 45 and every 60 degrees so  if we made a point right here at about 30 degrees again about 30 degrees up right here can label it 30 degrees and then if we think about it if we keep going by 30 degrees that's how forget our 60° and if we kept doing this you guys we would split the top half into 1 2 3 4 5 6 pieces so in radians we'd split this one pint is 6 pieces every 30 degrees is a pi/6 460 degrees you have to pi over six that we know two of her six to just be one third or pi/3 Here We have forever 6 part of me we have 3 or 6 or 3 or 6 and then with our next 30 degrees we have four over six that again we know how to reduce directions to we know that that would be two thirds pie and that would be at 120 degrees if we made an 30 degree point  that would be 150 degrees and if we kept going by those sixes we have one two three four five six hi  SMS just keep filling in our every 30 degrees and every 6th it's always fun to say 6 so list 30 degrees of how did 3280 we would get 210 degrees and then in radians that would be one more six-week of 7/6 pi and if we had a weather is 30 degrees we would have 240  and then that would be 8/6 but we remember how to reduce that would be 4/3 pi  every add another 30 degrees here we would be at 300 degrees and if we counted our sixth that would be 10 over 6 or 5 3 High and if we kept going this would be 330 degrees and then if we add it or 6 always so fun to do alright we would have 11/6 time we double-check to make sure I am correct in that fairly certain I am I am 11 / 6 great then we also have one right in the middle of all of these at every 45 degrees we're going to fill those and last right in between  every 45 we can see that we're also long nose up into fourth now if we split up this whole thing every 45 would happen every one two three four so you can make that pie over 4 radians  and over here we would have three fourths  3/4 radians and in degrees every 45 that would be 135 degrees down here another 45 degrees would give us 5/4 Pi radians and we would be at 225 degrees and again I'm getting this just by adding 45° everytime and counting by fourths Pastor here I would have 6/4 between I would be 3/2 I found another one I would have 7 / 4  hi and if I added my 45° between here and here I would get 315° alright so that is where all of the degree values are derived from and these four coordinates are derived from and that's also where all of our radian values come from there's no really special about them it's just how we break up the unit circle I would I have to show you next it is rather interesting and it isn't as obvious we fill in you guys know that there are coordinate values for every one of these points and we still doze and using our knowledge of cosine and sine trigonometric ratios of sine and cosine so right up here let's get a little here for ourselves about what those mean cosine  we remember is  the edge side over the hypotenuse  and sign we remember is the opposite side over the hypotenuse  and for the sake of the unit circle remember that sign values are your wise for every coordinate and your cosine values are your ex's for every coordinate  now remember what I said about the special right triangles how we were going to use them here they are special right triangles let's add another note how about right over here yes remember that we have to what cases of special right triangles we have  one that looks once side one long side and then we have an isosceles right triangle two sides of the same length  and the short side long side Triangle thats are 30 60 90 triangle 360 here's Thirty Andy's please have the same ratios of side lengths where this long Stein opposite hypotenuse that's the same as name a radical 3 the short side is one and the hypotenuse of the value of 2  445 45 triangle are same side same length sides are both radical 2 / 2 and hypotenuse is 1 alright I'm just realize that's pretty small here's a little preview of close up of what I just wrote  if you needed a little bit better to visual  text now we're going to use those to help us  we can create these triangles I'm going to use a pencil and I'm going to race we can create right here at our first point you guessed it it's written right there a 30 60  90 triangle how cool and we can use our cosine to get RX and can use their sign to get Ry so Arco sign is that recent over hypotenuse to here's the angle or using the adjacent side would be this guy right here and if we look at our reference special triangle we know the fat is going to be the square root of 3  and then least say over the hypotenuse and a hypotenuse of ours so 30 60 90 is too so their ego or x coordinate is square root of 3 over 2 now to get our y-coordinate did you sign the opposite side from this reference angle is are short and that are short and for a 30-60-90 is one play stay over the hypotenuse length and the hypotenuse length  and there we go that's how many find each and every coordinate do the next two for this quadrant  erase this in pencil so we can just keep creating these triangle and if you're wondering about what to write and how to show all of this work I'm not requiring that you draw all of these special right Legos with me but I just you expect to see you filled out unit circle on your notes alright so the next special we can see what kind of gave it away already to ourselves is it going to be a 45-45  right triangle we're going to use those properties to help us out to get our x value we're going to look at cosine we're going to look at the adjacent sides of this ankle which is this side right here square root of 2 over 2 we know that's going to be the top of our fraction we know we're going to have sqrt 2 / 2  / are hypotenuse value which for the special right triangle is one but we know that anything divided by one and doesn't actually change it so we're going to leave it as square root of 2 over 2  alright going to erase that real quick now to get the sign value for our wine do the same thing that we will get the opposite side while the opposite sides actually the same length special triangle so again we have squirt 2/2 / 1 or hypotenuse and we know anything / when does it changed at all so we have square to ever again  alright one last triangle to draw here erase these lines and a kind of already gave it away to ourselves cuz we already wrote Our ankles we are going to be doing a 30-60-90 triangle so here is our  triangle here is a reference angle we're going to take the cosine first to get her ex so we're going to look at the adjacent side to our 60 which is one over and then we're going to have  the hypotenuse which is the square root of 3 stop now if you'll remember this isn't how it looks on your unit circle because we have to rationalize a denominator member we can't have radicals in the denominator rationalize that by multiplying top and bottom by the square root of 3 oh my goodness I made a mistake today this should not be radical 3 we have to look at the adjacent add 1 and divide by the hypotenuse not radical 3 but to my goodness good thing I caught that so the x value hear your cosine would be 1/2 and then to get your y value your sign you have to take your reference angle 60 and you have to look at the opposite side in this case that would be radical 3  play put it over the hypotenuse value which is to and there we go that's how you create all of these coordinates I'm not going to keep going through and this whole circle and do special right triangles for all of these but see how it's created  and I'm not going to even require you to continue to fill out the coordinates for the rest of the circle just have to do these three because I already gave it a handout in class and this really isn't new information I just wanted to show you how we come about this Corvette okay moving on from our unit circle we're going to talk about the. a trig animatronic function now remember that the. Just means how long it take Spore the graph two repeat itself so what am I talking about saying the graph of sine a sine function of a period of a sine ocean is really just how long it takes for the sign values to repeat themselves so if we take a look at our unit circle we can see that we have sign I use for every quadrants  until two pie and then they start to repeat themselves right we have to go around the whole circle shoespie and then they repeat go to 2pi then we have the same ones go to to pie and then we have the same one  so we know that the set up a regular sine function is going to be too high because that's how long it takes for your values to turn around that Circle again  and we're going to demonstrate that graphically too so I'm going to show you on the calculator what I mean by that since really going to help illustrator calculator  and we go to y equals and we type in sinx I'll scan it out since I've been with you what is the sign which is down here  and then press X and it's right next to start all those variables and then you close your parentheses go to window and we're going to do something really neat here at the better illustrate that  components what's the sine function we're going to change our window so for X-Men we're going to say -2pi here's how you can do that you type in -2 and then you can press seconds and that little carry button it's right here below clear  arthropods and then  you can make your ex maximum values for pastor compressed for and then and that carrot button again  in our scale just a really sure what's going on here we're going to press second and then pie and we going to do play over to because remember if you look at your unit circle  Pi over 2  is where sign crosses zero  or part of me sign goes up to 1 x that's going to be signs relative Max and then if we go down to Ry minimum we're going to make it negative to Ry Max is going to be too and we're going to go by ones for Ry now we're going to grab this thing let's see what it looks like alright so it's showing up and here's what's really cool about the sine function as you can see RX can use are those radians right we made them Radiance when we just type in our window we did them every pi over 2 and so we can see that ever two of these every pie and every four of them is every two pie or every four of them is one rotation around that Circle so it's pretty cool stuff block it off  when I get my posted if we sort of block it off in take a look at what's going on for every two pie  bright this is one of them if we can look at every four we can see that this is what mine is doing in this makes sense because again if we look at our unit circle we know that sign starts at zero goes up to one goes down to zero goes to -1 and then goes back up to zero and by the time it's back to zero that's when it starts going around the circle again so that if you remember is how this trigonometric functions are graph the X values are the radians and the dependent Val card values what's whatever was in this coordinate right okay so using that kind of visualize here remember this piece of information from our last video when we had y equals a sign  as x minus t detox diet that is your amplitude also your vertical Strat drink  Envy was called your. But I don't really want you guys to think of it like that just yet it does have to do with your horizontal stretch and Shrink  but here's the thing your value actually comes from this ratio your back value actually comes from 2 pi over this absolute value of this be here's why remember when I showed you on the unit circle  that appears how long it takes for a graph to repeat itself and it takes stop sign a whole rotation of the circle repeat itself it takes a whole 2pi to repeat itself  so that being bed  when we have  Novi value or pardon you have eval one doesn't really show up then we have a regular rotation IV have a rotation of 2pi just like it normally would be but that's why thanks that Wendy is greater than one  we would plug that into here and we would get smaller value  we would get a shrink Telegraph if we made that be lower than one we have a decimal on the bottom we have a bigger number here for our overall. Value Anna not with stretch  are graph horizontally Isom holding a bat makes a little bit more sense. We reviewed the unit circle as well as the sine function  now there's one last topic I'd like to talk about or moving on and we're going to talk about properties Messick's inverse abbreviate trigonometric functions end I want to review 1 vocab term with you for this oscillation oscillation just games when a graph continually snooze up  and down and we saw that trigonometric functions always do because there is by nature a part of that unit circle and they just keep going around and around and around that Circle they go up and then go down and that's just that's how we graph trig functions an important thing to remember is that in order for a function to have an inverse it needs to be able to pass the horizontal line test and we know just by thinking about it but something that oscillates won't ever pass that hlc whenever pass the horizontal line test but we can do something with our ship functions and so we can do with me we're going to draw out sine function again  we're going to label  every negative one every positive one we're going to go by pi/2 on our x-axis again I'm just counting by halves here okay and if we remember how sine function behaves we have a beginning here at zero zero and then over to it comes all the way up to 1 a tray dealer Old Tie we have a value of 04 sign at 3 Pi / 2 we have a value of -1 for occupy we hit again at 5 / 2 pi we have A1 again and then at 3 pie we come back down to zero make sure you have arrows at the end of your graph always sister sine graph of you talked about this will never pass the HLT just by looking at it we know but what we can do with trig functions is pretty interesting if we break them up and take some posts to show you what I mean if we break them up into sections so say we broke him up the sign turn into every pi over 2  check out what just happened  for this for this one change of Pi from the range of pi/2 to 3 over 2 pi this little section of sine of x this does pass the HLT section for this domain restriction from pi/2 to 3/2 Pi we can in fact have an inverse and so that is where inverse functions come from the write this part down with me  sign next is 1 - 1 4 the interval or on the interval what we just sound pi/2 - 3 / - Pi + answers can hardness to see if you look these intervals can also change right it could be Pi / 2232 / Pi I could be 3 over 2 pi - 5 over 2 pi it has you have a lot of different options it's not just one domain restriction and so we know that it's one to one for this interval so that means that for that interval and all the other ones I just point it out arcsin  avexir still function right for me make sure you have your wine goes in there why you close arcsin or an inverse notation sine to the negative 1 of x does exist for the domain of -1  Chowan and the range of this interval negative O card me hi / 2 23 / 2pi  and remember that may look a little bit backwards but it should look backwards because it's the universe so what our domain was for the original function or member is a range for the inverse and what our range was for the original our domain for the inverse trigonometric functions cosine of x tangent of X what else do we have cotangent of secant of X cosecant of x lower face function do then have inverses if you restrict their domain so you just have to be careful about where restrictions are right you have to make sure that it will pass the HLT with an whatever interval is stated okay what was the last little piece of trig do I need to show yes it will see you in class tomorrow thanks for watching

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AP Calc Trig Review

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