Transcript for:
Inverse Trigonometric Functions Overview

[Music] thank you all right welcome back everybody uh today we're going to be doing section 6.1 so if you watched the um if you watched my lecture on Section 4.2 where I said we were going to be doing section 6.2 I misspoke 6.1 is what we're going to do next and this is the section where we learn about the inverse trigonometric functions um 6.2 is a continuation of the content about the inverse trigonometric functions but I decided to push that until after we do section 7.1 because I feel like most of the examples in that section are in section 6.2 are easier to do once you've had section 7.1 so we'll do that later for now let's get started with 6.1 and we're going to learn about the inverse sine cosine and tangent functions but also we're going to learn about the other ones the inverse cosecant arcs the end of the inverse cosecant the inverse is secant and the inverse oh geez geez I'm sorry the inverse cosecant the inverse secant and the inverse cotangent functions yes that's right I'm sure I'll remember once I look at the list again there we go um so first we need to talk about domains and ranges then we'll do the inverse sine function cosine function tangent function cosecant function secant function and cotangent function so the inverse of all of those things um I will say this this section is a little bit long so you might want to settle in you might want to take a break in the middle that's okay there's a lot to cover in just this one section so bear with me but I'll try to keep things brief if I can all right so first remember that a function has to be one to one in order to be invertible so we need that property to be true but the thing is the trigonometric functions are not one to one and so they don't have inverses outright they fail the horizontal line test and they fail the horizontal line test pretty pretty miserably right if you think about for example the sine function well if I try to draw a horizontal line through it it hits the graph many many times in fact it hits the graph infinitely many times so outright when you think about for example the sine function defined on the real line it's not invertible that way so what do we have to do we can make them invertible by restricting their domain and I mentioned this in that the lecture for Section 4.2 but we can talk about inverses of functions that aren't outright invertible if we can restrict their domain in a meaningful way and so that's what we're going to do in this section but I will give you this word of warning in general you have to be very careful and you have to give very careful consideration to the domains and ranges when inverse functions are involved so just be cautious think about what what are the inputs what are the outputs what are the domains what are the ranges because sometimes if you ignore those problems will arise as we'll see so let's talk about the restrictions that we're going to implement first so we are going to restrict the domains of sine cosine and tangent as follows so we're going to restrict sine to the angles sorry these should all say Theta is my bad this should be Theta right here I can change that later Theta Theta Theta there we go Theta oops Theta Theta so for the sine function what we're going to do is we're going to restrict our domain to angles that are within negative pi over 2 to pi over 2. so on in terms of the unit circle that's all the angles that are over here on the right hand side of the circle including pi over 2 and negative pi over 2. so remember what is what is sine right sine is vertical displacement so if you pick a point on this curve I'll go back to a black marker here there we go so if I pick a point sine is the vertical distance here the vertical displacement so one of the nice things about this Choice as we're going to see in the next slide I believe is that we will be able to recover all of the out all the possible outputs of the sine function using this domain restriction all right so sine is going to be restricted to negative pi over 2 to pi over 2. for cosine for cosine what we're going to do is we're going to restrict ourselves to the interval 0 to Pi 0 to Pi including 0 and including Pi so if we want to be able to talk about inverse cosine we have to restrict the cosine function to this domain if we don't we can't invert it can't talk about inverse functions and we're dead in the water so we need to include this restriction then lastly for the tangent function we're again going to restrict ourselves to the right hand side set of angles but now we're not going to going to include negative pi over 2 or pi over 2. and that's frankly just because remember if you have a vertical line then it has an undefined slope so we can't talk about tangent of pi over 2 and we can't talk about tangent of negative pi over 2 because the function is just not defined there all right so these are the unit circle representations of the restrictions that we're going to have for our functions so more specifically we have this here's the sine function and its restricted domain is negative pi over 2 to pi over 2 and its range is still negative one to one and then cosine we have 0 to Pi negative 1 to 1. and then for tangent we have negative pi over 2 to pi over 2 exclusive and then the range is negative Infinity to Infinity all right now one thing that's important to note here is that each value in the interval negative one to one is actually still hit with these restrictions on sine and cosine and this is what we want we want this because what that means is even though we have cut most of the domain of sine and cosine away we can still achieve all of the possible outputs of those functions just on these intervals here which is great because now we're not we're not losing any information and this is largely because of the periodicity of these functions right they're periodic so they repeat and as long as you restrict the domain to a point where you can get all those possible outputs then everything is good everything's covered um the same thing for tangent so even even though we restrict tangent to B in the interval negative pi over 2 to pi over 2 exclusive the range is still going to be negative Infinity to Infinity so we're going to get all the possible slopes of these lines so we've lost no information in terms of outputs which is great all right so we have this since the inverse functions or since inverse functions invert the domain and the range we have these this set up here so sine inverse otherwise known as arc sine has domain negative one to one inclusive and has range negative pi over 2 to pi over 2. so yeah remember that inverse functions invert the domain and the range they switch the domain and the range so when we want to talk about the inverse sine function the domain and the range are switched uh I'll say this too I put both of these symbols on here because these two symbols are interchangeable and I have a note about them on the next slide yeah on the next slide but you can use these either one of these symbols they're acceptable everywhere uh sine inverse or arc sign cosine inverse or Arc cosine tangent inverse or arc tangent arc tangent uh yeah totally interchangeable symbols um I will use them interchangeably sometimes I prefer tangent inverse sometimes I prefer arc tangent it really just depends all right uh so you might be asking yourself why are these functions called Arc dot dot dot well the main reason is that the output of these functions is an angle the output of these inverse trig functions is an angle so that's the one takeaway that you need to like cement in your head especially what's down here in the box but I'll get to that in a second when we work with the inverse trig functions what we're doing is we're putting in an output of the original functions and we're getting out an angle that would lead us there but remember that we're using radians right we we do all of our stuff in radians and remember that radians are the lengths of Circ of circular arcs on the unit circle so that's why they're actually called arc sine and Arc cosine and arc tangent and Arc cosecant and Arc secant and Arc cotangent because what it's giving you is it's giving you the length of that circular Arc it's giving you that angle in radians which is really cool so maybe now you might feel tempted to always call it arc sine Arc cosine and so on um so a lot of times students will tell me that they struggle with inverse functions and what I try to do to get them through that struggle is they should always rephrase inverse functions in these terms so when you look at a function like arc sine like for example if I said oh what's um what's arc sine of one right the way that you answer that question is quite literally by saying this sentence in your head or out loud even out loud right so um what is arc sine of one that is the angle that has a sine value of one hmm what angle has a sine value of one think about it for a minute my angle has a sine value of one what could it be it's pi over 2 right it's going to be pi over 2. pi over 2. oops it's kind of a funky looking pie here we go there you go it's pi over 2. how did I figure that out well I imagine the unit circle and I said this sentence out loud to myself up what angle has a sine of one and then you think about the unit circle and you think when the sine value is going to be 1 and you say oh it's angle pi over 2 right uh yeah so that's what it is ask yourself that question this question that's in the box that I circled that I'm pointing at memorize that sentence say that to yourself whenever you're trying to do anything involving inverse trig functions and frankly just inverse functions period um okay so let's focus on the inverse sine function for a bit so how do we Define it well the inverse sine function or the arc sine function is defined as y equals sine inverse of X or arc sine of x if and only if x equals sine of Y one thing that I will say or yeah one thing I'll say here is don't don't let the x's and y's throw you off or confuse you you always want to just think of things in terms of inputs and outputs remember that these letters X and Y they're totally arbitrary they don't have to be X's they don't have to be y's they could be s's they could be T's they could be smiley faces they could be laughing emojis like those symbols can be anything it doesn't matter that one says X and one says why all that matters is the relationship between those symbols related by the functions sine and arc sine that's all that matters um so what is arc sine of x well arc sine of x is going to be the angle whose sine is X and that's what it is so when you're like we just did a moment ago right when you're trying to evaluate arc sine you just say what angle has this sine value the sine value that I'm using as an input right okay so notice here the inputs of these functions are the outputs of the original functions and so what these inverse functions tell us is the angles that would lead us to those outputs all right let's look at a picture here so this is a graph of the sine curve as it's defined on the whole real line and again it totally fails the horizontal line test and you can see it you can see it because if you if you draw a horizontal line it's going to intersect the graph many many times so we needed that restricted domain and that's actually what's pictured right here so this is the restricted part of the sine function that we're going to use to be able to define the inverses so here it is pulled out and now we can look at it more directly so if we want to be able to talk about arc sine or inverse sine then we must only consider this part of the sine function from negative pi over 2 to pi over 2 inclusive inclusive and if you remember how to graph inverse functions the graph of an inverse function is the graph of the original function flipped about the line Y equals X so this is what arcsine looks like let me highlight it here there we go I highlighted it in an orange right there so that's the graph of arc sine that is like if you wanted to actually draw the picture of it that's what the picture looks like and that's it that's it that's the whole graph right there how did you how do you get it well you take the graph of sine which is right here and you flip it around the diagonal and then you get the graph of arc sine all right now let's try an example here so let's do a couple of these um and I want you to try it on your own and I want you to try saying that phrase out loud just a reminder since we're working with arcsine this is the phrase let me erase this other stuff yeah that's the phrase so for this next next example this is what I want you to say out loud to yourself when you're trying to work on this problem all right so let's do the first one let's find the exact value of sine inverse of 1 and then sine inverse of negative one-half and then arc sine of square root 3 over 2. so think about it for a minute say it out loud and try it on your own thank you all right now let's do it together uh the first one we actually already did uh what is sine inverse of one that's the angle in negative pi over 2 to pi over 2 whose sine is one so in other words it's the angle Theta the angle Theta such that sine of theta equals one well since sine is vertical displacement you imagine the unit circle and you think about where that vertical displacement is going to be one and then you can clearly see that it's going to be pi over 2. that's the angle that will give you a sine value of 1. so arc sine of one is pi over 2. what about the next one what's a arc sine of negative one-half well arc sine of negative one half is the angle whose sine is negative one-half the angle such that sine of theta is one half so how do you do this well you imagine the unit circle again right here's the unit circle and I need the sine value to be negative one-half so it's a negative angle the displacement is one-half so we get the short leg here and what's that angle going to be that angle is going to be negative pi over six and so that's our answer our answer is negative pi over six because sine of negative pi over 6 is equal to negative one-half there we go and that's pretty much it um all right now let's do the last one what's arc sine of root 3 over 2 well it's the angle whose sine is root 3 over 2. so what is an angle within our restricted domain that has a sine value of root 3 over 2. so what do you do you imagine the unit circle right you think about the unit circle and you say Okay I need the sine value to be root 3 over 2 root 3 over 2 is the long leg so it's going to look like this and so that tells us the angle is going to be pi over 3. and voila there it is oh I'm sorry I'm sorry uh I realize I have a typo here this should be positive so I probably scratch that little bit out there I'll fix that later scratch that bit out positive root 3 over 2. and you can think about it if you want an extra exercise what would be arc sine of negative root three over two you can probably figure that one out in just a couple minutes all right so let's talk about the cancellation equations so one of the things that I mentioned in that lecture 4.2 is the cancellation equations everybody loves these this is one of the things that we use these inverse functions for but if you do the inverse so if you if you do sine of ah man if you do sine of a value and then you take the arc sine of that value you're going to end up with where you where you started with the original input and vice versa if you do sine inverse of X and then you do sine of that you're going to end up with X again so in essence arc sine and sine cancel each other out but it's it's not quite that way so that's why I have this important note this important note down here when you're evaluating these Expressions it is very important to pay careful attention to the domains and the ranges of the functions involved because the cancellation will not always work out the way that you expect it to so you're not always going to end up with X what I mean what I mean is you're not always going to end up with the same thing that you started with because the domains that we restricted things to play an important role we'll see in just a minute here okay so let's try this one um try to do this one on your own I want you to think about it uh think about it on your own and then I want you to say some stuff out loud and we'll do it together in just a minute all right okay so now let's try to do it together and first I'm just going to try to make sense of what this is even asking like what is this even asking me um so this is saying find the exact value of sine inverse of sine of pi over 8. sine of pi over 8. now the first thing I notice is pi over 8 is not one of the standard angles that we know so what is sine of pi over a well I know that that's going to be some weird irrational number but I know it's not going to be like root 3 over 2 or 1 half or root two over two or one or zero it's not going to be any of those so if I wanted to see what this is I'd either have to use some sort of formula like we'll learn later in the course or I would have to use a decimal approximation from a calculator but that doesn't mean that I can't solve this or it doesn't mean that I can't compute the exact value of this expression because what we're doing here is we're doing sine inverse of sine of pi over 8. so we're going to look at the vertical displacement that you would get for sine of pi over 8 and then we're going to see what the angle would be that would give you that but the angle has to be within the restricted domain so let's take a look at this um oh yeah there we go I just cut right to it so Pi of Pi of sorry pi over 8 is within our restricted domain negative pi over 2 to pi over 2. so that means that the cancellation equations apply they apply outright these things cancel each other out and you're left with pi over 8. there you go all right now let's try something a little bit trickier here we go so try to do this one on your own give it some extra pause and then we'll we'll do it together in just a moment all right let's try to do it together um what is sine of 5 pi over eight uh well that's another one where it's not a standard angle so I don't have an immediate answer to that but the other thing is that 5 pi over eight I can kind of visualize where that is right let me do this let me draw the unit circle let's see 5 pi over 8 that's 5 8 of a pie so 5 8 of a pi is probably going to be like around here just a random guess because if I were to cut this into a four pieces like so that would be five of them oops just erased it there we go so this angle right here is 5 pi over eight and so I can see that there is a sine value there but here's the catch I can't apply the cancellation equations because 5 pi over 8 is not in the interval negative pi over 2 to pi over 2. so this thing is not going to be equal Excuse Me 2 5 pi over 8. it's not going to be equal to that but we can still evaluate this because remember sine of 5 pi over 8 this piece is just the vertical displacement right it's the vertical displacement that you get when you trace out that angle 5 pi over eight but there is an equivalent vertical displacement that does correspond to an angle that lies within our restricted domain and I'm going to draw the picture right here and then I have a better one on the next slide but we can look at this angle instead sorry it's a rough sketch a rough sketch but look this vertical displacement is the same let me Point here this vertical displacement is the same as this vertical displacement that really just has to do with the way that sine is defined and the Symmetry involved with sine right so what is this angle here well this angle is going to be 3 pi over 8 or 3 8 of a pi so let me go to the better picture and we can make more sense of it um yeah so this picture is much nicer but you can see that this angle right here is 5 pi over 8. and that gives us this vertical displacement but that vertical displacement is identical to this vertical displacement right here and this angle is the angle 3 pi over 8. 3 pi over 8. so we have this result more specifically sine of 5 pi over 8 is equal to sine of 3 pi over 8. and now that we're looking at sine of 3 pi over 8 3 pi over 8 is within our restricted domain so we can actually apply the cancellation equations and we'll see that this is going to be 3 pi over eight there we go so it's a it's a little bit funky at first to try to get used to what's going on here and the other thing I want to point out is we I'm not saying that 5 pi over 8 is equal to 3 pi over 8. I'm not saying that those two angles are equal what I'm saying is the vertical displacement oops the the vertical displacement for those two angles is the same and vertical displacement is sine so it's not that the angles are the same it's that the output of those two angles under the sine function is the same so written out nice and clean here we get that sine inverse of sine Phi sine of 5 pi over 8 is equal to sine inverse of sine 3 pi over 8 and now we can apply the cancellation equations and we get 3 pi over 8. there we go we're going to do another another couple examples like this as well because it does take some time to get used to it my best advice right now is remember the unit circle everything goes back to the unit circle over and over again it's always like Imagining the unit circle remembering what sine is remembering what cosine is remembering what the angles are the standard angles and everything just comes out of that there's surprisingly very little calculation that's going on here it's more just interpreting the unit circle all right so now let's move on to cosine the inverse cosine function or Arc cosine function is defined in this way we say Y is equal to Arc cosine of x if and only if x is equal to cosine of Y more intuitively if you want to work with Arc cosine this is how it's defined Arc cosine of x is the angle whose cosine is X so just like before except now we're doing cosine all right let's look at the graph once again outright the cosine function is not invertible it's not invertible it doesn't pass the horizontal line test so we needed to restrict the domain and when we restricted the domain we got this here we go here's what it looks like when you peel everything away and that's the graph of cosine on our restricted domain from 0 to PI right 0 to Pi inclusive so what is the graph of Arc cosine look like well I think it looks a lot more fun than the sine function or the arc sine I'll say but how do you get it you're just going to reflect it about the line Y equals X and when you reflect that about the line Y equals X you get this so this is the graph of Arc cosine there you go so you can look at it for a minute ponder it for a bit looks a little funky but that is what the graph is that is the graph of Arc cosine all right now let's try some examples uh the flavor of these examples is identical to what we saw previously if you want to be able to do problems involving Arc cosine you just have to ask yourself what is the angle that has this cosine value so think about this one on your own say that phrase out loud and then we'll do this together in just a moment all right now let's do it together what is Arc cosine of zero well that's the angle whose cosine is zero what angle has a cosine of zero I gotta think about it okay let me draw the the unit circle the cosine or cosine is horizontal displacement right so what angle has zero horizontal displacement oh it's going to be pi over 2. there we go so right off the bat I can see that Arc cosine of 0 is going to be pi over 2 Y is that specifically because cosine of pi over 2 is 0. so there we go there's the answer now let's try to do the other one let's try to do the other one what is Arc cosine of negative root 2 over 2. that is the angle whose cosine is negative root 2 over 2. let's try to draw a picture of that angle here's the unit circle negative root 2 over 2. that's one of those diagonals right that's one of the standard values and it's negative so the angle is going to go down and this is the angle we're looking for right but since it's going to be negative root 2 over 2 we're looking at the horizontal displacement here right cosine is a this is negative sorry sorry I apologize I'm thinking of vertical displacement see this is why you got to draw the unit circle each time we're looking at horizontal displacement horizontal displacement so when is the cosine value going to be negative root 2 over 2 when it's over here oops not the Eraser come back there we go right so this value right here is negative root two over two and this angle right here is going to be what three fourths of a pie so 3 pi over 4 yeah 3 4 pi and that's what it is so what is Arc cosine of negative root 2 over 2 it's 3 pi over 4. oh there we go 3 pi over 4. and that's the idea um yeah I mean that's pretty much it so once again uh there's actually very little calculation there's not really any algebra that you're doing you're just thinking about the unit circle thinking about where these values lie in terms of Sines and cosines and tangents and such and then asking what angle would lead you there what angle would lead you there all right next just like with the sine function there are cancellation equations for cosine but the catch is that those angles have to be within the restricted domain so if you want to be able to apply this result your angle has to lie in the interval 0 to Pi and if it doesn't you cannot apply that result and so that's why I was saying earlier that sometimes this value will not just be X again it won't just be X again it really really depends on what the angle is and where that angle lies so let's try this example what's the exact value of cosine inverse of cosine pi over 12. try this one on your own think about it say say it out loud say that phrase out loud and see if you can figure it out all right let's do this one together um let's see pi over 12 that is not one of those standard angles so I don't know what cosine of pi over 12 is outright I couldn't just say oh cosine of pi over 12 is blah blah blah blah all I know is that it's cosine of pi over 12 but what I do know is that pi over 12 is in our restricted domain it is in our restricted domain so that means that I can apply the cancellation equations these are going to quote unquote cancel and so we'll be left with pi over 12. there it is all right so it's nice and it's nice when the when the angles are in just the right place but let's mess things up a little bit here let's try this one all right I want you to try this one on your own at home and then we'll do it together in just a minute remember to say the phrase out loud all right let's try it together um let's see cosine of negative 2 pi over 3 ah 2 2 pi over 3 that is one of the standard angles so maybe I can compute this in my head right okay so cosine of negative 2 pi over 3 oh what is that um that's going to be negative obviously I have to draw the picture I have to draw the picture negative 2 pi over 3 is over here right over here so cosine is going to be negative one-half right negative one-half that's what it is that's what it is there we go there we go um okay so then what is what what is uh an angle that has a cosine of negative one-half well we know that negative two pi over three does so maybe that's the answer but wait that's not the answer that's not the answer because negative 2 pi over 3 does not lie in the restricted domain it doesn't lie in the interval 0 to Pi so the cancellation the cancellation equation cannot be applied here we can't apply it so what we have to do is we have to think about the unit circle and we have to think about the restricted domain and then we have to think about what angle in our restricted domain is going to have a cosine of negative one-half and if you look at the picture I've got down here it's going to be right here there you go I'll try to draw it like so it's this angle right so instead of going down this way we have to go up this way because this is how we restricted cosine right we restricted cosine to the upper half of the unit circle and once this angle this angle is actually two pi over 3. so that means that this expression here the exact value of arc cosine of cosine of negative 2 pi over 3. is actually 2 pi over 3 positive yeah there we go there's the final answer here I'll go back for just a minute here um this was essentially what we were figuring out with the unit circle picture I drew cosine of negative 2 pi over 3 is equal to cosine of 2 pi over 3 right so we can essentially replace cosine of negative 2 pi over 3 with cosine of 2 pi over 3 and then we can apply the cancellation equation all right okay so that's it for Arc cosine or inverse cosine now let's move on to the inverse tangent function so we're about almost halfway through what's the inverse tangent function or the arc tangent function arc tangent is defined as y equals Arc tan of x if and only if x equals tangent of Y tangent of Y and oh I realize I have another typo here yikes so maybe you can pick it out before I get to it we need our x values to lie in the real numbers but we need our y values to lie in this interval but these should be parentheses these should be parentheses why is that well we can't do tangent of pi over 2. it's undefined there so that's a typo that's an error those should be parentheses not brackets um but again what's the intuition for arc tangent how do you actually compute the arc tangent of something this phrase you say this phrase out loud arc tangent of X is the angle whose tangent is X then you got to think about the restricted domain then you got to think about that okay so let's take a look at this uh here's the graph of the tangent function itself remember tangent is not invertible outright it does not pass the horizontal line test so that's why we had to restrict the domain um how do we restrict the domain we're going to restrict it to the the region negative pi over 2 to pi over 2. exclusive just want to emphasize that again that it's exclusive there you go and this is all we need because again we'll get all of the possible outputs from negative Infinity to Infinity exclusive as long as we just look at this part of the graph um I realized I didn't say it earlier but I should say it now um the choices of restriction that we have are somewhat arbitrary what that means is if tangent is a good example it's easier to see there's there's no law that says that we must restrict to this region in here there's no like law that says we have to do it that way it's really just a convention to make things easier on us we could have just as well chosen this restriction instead we could have said oh why don't we pick uh this interval instead and just look at the tangent function on the interval negative 5 pi over 2 to negative 3 pi over 2 and all of the results we have would follow we would still get the outputs everything would essentially be the same the only difference is that we have to think carefully about where the angles lie when we try to apply the cancellation equations and when we try to evaluate these Arc tangents and so on so just to keep things simple we as in like the mathematical Community decided let's just pick the one that seems the most straightforward negative pi over 2 to pi over 2. and then for cosine it was 0 to PI right okay so if this is what the tangent function looks like on its restricted domain if we want to see what the graph of arctangent looks like we reflect it over the line Y equals X and get my black pen here right reflected over the line ah Y equals X and we'll get this there we go and there are still asymptotes so what's nice about this function the there are still asymptotes at pi over two let me make that a little darker and negative pi over 2. so arc tangent is bounded and I have this really really important note so when you get to calc 2 in particular just remember we really really like arc tangent and we really really like cosine and we really really like sine because they are bounded they are bounded functions so arctangent is bounded to arc tangent is bounded between negative pi over 2 and pi over two that's what this picture is showing right here right and it's a little bit hard to get into but the reason that we like these functions because they're bounded is because they don't blow up anywhere they don't get too big they kind of stay roughly I mean they do stay exactly within this range negative pi over 2 to pi over 2 for arc tangent and then negative one to one for sine and cosine and that helps out a lot because when functions blow up to infinity or when they get very large uh it can cause some problems so we like bounded functions they're always very nice okay let's try this one so let's find the exact value of arc tangent of one and arc tangent of negative root 3. try this one on your own remember to say the phrase in your head and say the phrase out loud and then we'll do it together in just a moment all right let's do it together what is arc tangent of one that is the angle that has a tangent of one all right what angle has a tangent of one hmm tan theta equals one implies theta equals what well remember tangent is just the slope of that Ray the slope of that line right so if the slope is one then we saw earlier that that means the the angle has to be pi over four the angle is going to be pi over 4. so that tells us that arc tangent of one is pi over 4. Boop there it is not too bad right now let's try to do the other one arc tangent of negative square root of three this one's a little bit weirder because I'm like well negative square root of 3 what angle has a tangent of negative square root of three well this is where those standard angles come in handy and this is where the definition of tangent comes in handy because tangent is just sine over cosine right so uh this is often what you'll have to do for a problem like this we can rewrite negative root 3 as negative root three over two over one half let me use the highlighter for that actually right the the twos in those denominators will cancel and you'd be left with negative root 3. you might even want to think about it as negative root three over one for example well negative root 3 over 2 is sine of negative pi over three and one half is cosine of negative pi over 3. but sine over cosine is tangent and so what we get through this equation the string of equalities is that negative root 3 is tangent of negative pi over 3. just like so there we go so well if uh if if tangent of negative pi over 3 is negative root 3 then that means that arc tangent arc tangent of negative root 3 this Pi is negative pi over three there it is so one one thing I want to re-emphasize here is that to compute these things you really are just thinking of the stuff you already know you're thinking what is tangent what is sine what is cosine you're thinking about the unit circle you're thinking about these angles and the standard angles so it really does build on everything you already know not as much calculation as you might expect there is one little subtle point I'll point out for this question which is this remember the restricted domain oh see there's another one oh I got to go through and fix these I apologize those should be parentheses those should be parentheses um we need an angle between negative pi over 2 and pi over 2 that has this tangent value and that's actually why I had to choose negative pi over 3. right negative pi over 3. um if if the tangent is negative root three think about the unit circle draw a little sketch of it here remember tangent is just the slope so if the slope is negative root 3 it's going to look some the line is going to look something like this right so you might be tempted to say that oh it's this angle right here but wait that angle does not lie in this interval so we can't use that angle we must use this angle instead and what is this angle negative pi over 3 so that's why you're essentially forced to pick that and that was by Design that's by Design that's why we restricted the domain I mean that's because we restricted the domain all right just like before you see now I sound like a broken record oh man I have some bad copy pasting errors here I apologize these should be parentheses I'll fix those in the slides right after this um but again we have cancellation equations for tangent you just have to be very very cognizant of what's the input what's the output do these values lie within our restricted domain or do they not and that's what you have to do all right so ah okay yeah so since we've done a bunch of examples like this already I'll leave you to do some on your own if you would like the book has many many more if you would like to try that you will get to try many of them on your homework as well but again the main thing I can say is just say those phrases out loud so for tangent it was where to go this one right arc tangent is the angle whose tangent is X where's cosine come here cosine there you go Arc cosine is the angle whose cosine is X and then all the way back here arc sine is the angle whose sine is X and that's how you do all these types of problems okay so let me get back to the bottom here we're actually nearly done we're nearly done so there are three other inverse trig functions there's the inverse cosecant the inverse secant and the inverse cotangent um but they're pretty much the same thing they're they're pretty much the same process that we just did so I didn't include any other examples I think you'll be able to do these ones on your own we'll just go through the definitions here the inverse cosecant function or Arc secant is defined as y equals Arc cosecant if and only if x equals cosecant of Y now you just have to remember what the Restriction is so here is going to be our restricted domain for uh for cosecant negative pi over 2 to pi over 2 inclusive but we're also going to cut out the point zero gotta cut out the point zero um and again that's because you can't really you can't I mean not really you can't take the cosecant of zero because you'll get an undefined value you'll just get no value because it's undefined um but notice that that choice of restricted domain is is basically the same one that we had for sine right so remember the restricted domain for sine was negative pi over 2 to pi over 2. so the restricted domain for cosecant is also going to be negative pi over 2 to pi over 2 with the additional restriction can't be zero so we're cutting out the angle zero and how do you answer these questions intuitively the same way we've been doing you say well Arc secant arc Arc cosecant oh man Arc cosecant is the angle whose cosecant is X and you just work it backward all right similarly for the secant function everything is the same idea again so y equals Arc secant of x if and only if x equals secant of Y what's the Restriction the Restriction is the same as it is for cosine except we also have to cut out pi over 2. we have to cut out pi over two and then how do you answer these types of questions you say this out loud to yourself you say Arc secant is the angle whose secant is X and you work it out that way and then last but not least Arc cotangent or the inverse cotangent function how do you define it y equals R cotangent of X means x equals cotangent of Y and then this one is a little bit different so for this one well before I get to that the the domain is restricted a little bit differently it's restricted to zero to Pi zero to Pi but how you answer these questions intuitively is just this again R cotangent of X is the angle whose cotangent is X so seems a little bit funky um but that's it so what I recommend now go try to do the homework um just working with these inverse functions try those examples you can read more about them in the book If you would like and then like I had mentioned before the next section 6.2 I'm actually pushing to later in the course so that's going to occur after we do section 7.1 because those problems will be easier to answer once we have section 7.1 but on that note that's it goodbye everybody and I'll see you again next time thank you foreign