 Page 235 any questions from the homework you want me to go over the homework that I said was hugely crucial and important Now I gave you a lot. Hopefully you found though it started to get fairly Repetitive or fairly mundane. I hope so. I think I said to you last day my goal was that it becomes Boring for you and or you then that's that's where we should be at Any of these any of these any of these? Yeah, I Heard G and H 3 G and H left to 3g 180 degrees right there What's the reference angle do you think? zero Yep H negative 270 degrees You know right there, what's the reference angle? No, remember reference angle is either measured there or there in this case because it's the same distance either direction, but 90 it's always measured to the closest x-axis in this case both axes are the same distance away, but has a reference angle of 9 Weird ones, I know we don't use those ones very up. Is that okay? excellent Any others Then turn to lesson three Combined today With Monday's lesson and I said to you that that really for the next Three well for the next month and a half I'm not going to be teaching you anything new after that I'm going to be simply showing you different ways to use what you learned Two days two classes ago last class today and Monday today is Applications of reference angles and the cast rule and for those of you that were away last day Here is a quick explanation of what we meant by reference angles We said that the reference angle if you were in this quadrant that was your reference angle If you were in this quadrant that was the reference angle If you were in this quadrant that was the reference angle if you were in this quadrant That was the reference angle you guys have lines there. Do you not? Yeah, for some reason mine vanished and there is no hard rule to calculate the reference angle if you're in this quadrant The reference angle is your terminal angle Here in this quadrant. It's a 180 minus the reference angle gives you the rotation angle or here it's rotation angle is 180 plus the reference angle here it's 360 minus the reference angle gives you the rotation angle I actually don't memorize those Holly when I do the sketch I can just eyeball it and realize oh I'm here it's gonna be the that 360 minus gives me that okay so it says memorize I can figure it out by drawing a picture we also introduced the cast rule cosine is positive all of them are positive only sign is positive along with cosecant and tangent and cotangent are positive oh and I guess secant is positive down there cuz secant goes with cosine where do we use this why do we need this why am I doing such a big song and dance about this turn the page and get out your calculator on your calculator can you find for me the tangent of 40 degrees and give it to me to five decimal places please Amanda would you get point eight three nine zero nine is that right is that right I've seen people look puzzled yes no what's after the nine oh can you round off properly Amanda my physics kids we're all going round off properly is it is it that yes okay now that also means if I told you that the tangent of some mystery angle was point eight three nine one zero you should be able to find the angle by going second function point eight three nine one zero can you try that does that give you your 40 degrees close we've rounded off it's not exactly 40 but close yes if I say the cosine of 82 degrees is can you find the cosine of 82 degrees to five decimal places Andrew would you get point one three nine two to five decimal places that should mean that if I told you that the cosine of some angle was point one three nine one seven you should be able to go shift cos of that and does that give you 82 degrees back yeah Katie can you find for me the sign of 160 degrees to five decimal places please I heard point three four two zero two and of course that means that if I tell you that the sign of a mystery angle is point four three two zero two that should absolutely give you that hundred and sixty degrees back yes oh does it no wait a minute mr. do it when I go shift sign of point three four two oh two doesn't give me a hundred and sixty degrees back it gives me twenty degrees back by the way the reason is you see there's two angles that have a sign of point three four two zero two and as a matter of fact there's two angles that have a cosine of point one three nine seven and there's two angles that have a tangent of point eight three nine one zero and your calculator Cassandra is programmed to only give you the ones between zero and ninety and it was never an issue in grade nine because when you're dealing with triangles your angles were all between the zero and ninety just like that was also meant they were the reference angle we need to tell you all of this and here's what we're going to ask ourselves what if I tell you that the cosine of theta is point four five six three two nine there are two angles that have that as a cosine and my calculator can't tell me the second one the cast rule and reference angles can't this is what we're going to deal with we call that a trig equation when I give you an equation where the variable is an angle we call that a trig equation and we're going to start to solve those today here's what it says to solve a trig equation step one is to determine the quadrants the angle will be in by looking at the sign of the ratio actually what I'd like you to write next to step step one in capital letters we're going to use the cast rule step oh in the heading here it says solving equations involving sine or cosine the method I'm going to show you works for all of them which is nice and it's actually a fairly clean easy method so step one use the cast rule it's much easier instead of reading this they're a little bit verbally expensive a lot but too much English easier to do it example one the equation they want me to solve is this sign of mystery angle equals point five ah but they have to add a domain restriction because when I said that there's two angles that have the same sign or code section actually lied you could keep going around and around and around and around you keep finding co-terminal angles there's an infinite number of angles that actually satisfies this equation so what we're going to do from now on the TS is we're going to give you a domain restriction and usually it's going to be between zero and 360 usually we're going to say go once around the circle and quit every once in a while Jessica they'll change the domain so keep your eyes open but 99% of the time once around the circle and quit first thing I'm going to do I said is write the cast rule C A S G Jessica which trig function did they give me in this question sign and according to this question was my answer positive or negative positive where is sign positive what two quadrants I know that the answer is here or here actually both both the answers are in those two quadrants so I'll draw that and if this was multiple choice and there was any answers that had angles bigger than 180 I would cross out those answers that's step one the cast rule step two says find the reference angle here's how we're going to find the reference angle remember the abbreviation I used last day for reference angle there's my abbreviation for reference angle our theta and that's going to be on my calculator the inverse sign of point five when you go inverse sign shift sign of point five on your calculator what do you get sorry Amanda 30 Amanda that's telling you the reference angle that's telling you this angle here and this angle here because that's how reference angles were defined there's going to be two X answers I'm going to call them X one and X two and the first answer starts from the positive X axis and goes to my first drawing Andrew how big is that have to be one of your answers is 30 degrees Shannon my second answer starts here again and goes right to there how big does that have to be well how big is my reference angle Shannon 30 and how far all the way to here how uh-huh so if that's 30 and that's 180 how big in your head yes by the way that's your calculators can you find the sign of 30 is it point five very quickly can you find the sign of 150 is it also point five and that's the only way to find that second answer find the reference angle and use the cast rule to figure out what quadrant we're in oh that's what step three is saying determine the rotation angles using the reference angles in the bottom lots of writing easier to just do example 2a first thing I'm going to do is draw the cast rule C a s t it's a which trig function did they give me was signed did the answer work out to be negative or positive negative so where is sign negative here and here oh and if this is multiple choice and there was any answers less than 180 I crossed those out you will find sometimes Sabrina on some questions you may be done you may notice that there's only one answer that has one angle between 180 and 270 and one angle between 270 and 360 you may notice there's only one quit has to be that I almost always won't give you one like that because that's a bad question in my mind every once in a while those have slipped through on the provincial I need to find the reference angle the reference angle is going to be the inverse sign up and here is the key here is the key we already used the negative to figure out where we are we do not include the negative to find the reference angle we do not put the negative right there we find the positive decimal shifted sign always we're not going to use the negative in this one to find the reference angle we're not going to use the negative in this one to find the reference angle we're going to use the negative to figure out what quadrant we're in what is the reference angle for point eight zero nine zero step oh I thought I heard you say it which one 53 point nine nine would round off to what Stephanie 54 it does say to the nearest whole number so the reference angle is 54 degrees and Steph reference angles are always to the x axis it's telling me this angle and this angle are 54 here's the first angle starting from the positive x axis right to there Carson how can I calculate it now he said 180 plus reference angle we're going to write that in your homework if you want to go straight to the answer go ahead in our notes let's write down that we went 180 plus 54 so that when you're studying you know what the heck we did and can you in your head go 180 plus 54 I think you can 234 Carson the second angle is you that big how could I calculate how big that one is 360 minus 54 and I even would argue you can do that in your head Carson 306 let's check can you have find the cosine of 234 can you guys try the cosine of 306 and we should get very close to negative point eight zero nine zero maybe a bit off because we did some rounding is it right you really should have been done okay you guys check you guys yes does work anybody give me actually tried it you actually did try or you just nodding your head hoping nobody knows he didn't type it in you try it no come on check it you guys cosine 306 we need to find out those works yes a bit off but yeah and course of 234 yes is very close to negative point eight zero nine zero you know first thing I'm going to do is Castro C A S T Shannon which trig function to give me in this question negative or positive according to this question where is cost negative here and here right it's fine the reference angle now the reference angle is the inverse cosine of not negative point eight zero nine zero just point eight zero nine zero what do you get for the reference angle 36 and again it's all that's telling us that those two angles are 36 degrees because reference angles are always measured to the closest x axis how big is theta 1 how can I calculate it how can I calculate it 180 minus 36 I'll do that in my head 180 minus 30 is 150 minus say 144 yes how big is theta 2 theta 2 is that big it's gonna be 180 plus 36 216 also works for candid try this one on your own I won't freeze the screen if you get stuck you can look up I'll just do this but see if you can try this one on your own no quick and easy way to find both answers without reference angles and cast rule there are other ways but this is the second quickest but least memorizing stuff way the best bang for your buck there is a quicker way you can memorize there's actually formulas or if you know one sign you can find the other one if you know one coast you can find the other coast with you know one tan you can find the other tan by subtracting or adding multiples of pi to save one whole second not worth memorizing three new formulas how many got that okay excellent here's the nice thing not only does this work for sign and for cosine and for tangent this same method with one very minor tweak will work for turn the page reciprocal trig functions like cotangent or cosecant so example 3 find the measure of angle a and once again we're between 0 and 360 cotangent goes with which trig function this is the same as 1 over tan a equals 0.5 I'm actually not going to teach you how to solve cotangent or cosecant or secant all I'm going to teach you to do is how to change it into sine or cosine or tangent why why learn something new when I can turn it into what I already know now to solve and tidy this up I would cross multiply don't write this next bit down I'm going to do this the long way and then I'm going to show you an easier shortcut that is worth memorizing so if I cross multiplied I would get 1 times 1 equals 0.5 times tangent yes and then how to get the tan by itself here is the easier way if they give you a reciprocal trig function but he is flip this and flip this at the same time in other words that's replace it with tangent replace that with 1 over it yeah what trig function did they give us here cosecant goes with what sine replace this with sine replace this with 1 over negative 2.86 that's the quick and easy way instead of cross multiplying and three lines later ending up there I'm saying for what it's worth flip both sides you flip the left side it becomes a nice trig function flip the right side just becomes one over whatever you have easy that's what we're gonna do so I'm going to erase this this this and this and I'm going to say hey this is really just now I know 10 sorry I know 1 over point 5 is to who cares Amanda if it works out evenly I'll just leave it as a fraction anyways because most of the time it won't work out evenly and I'm going to go to my calculator what trig function did they give me cotan not cotangent what trig function did they really give me tantan okay casserole C A S T Roxanne according to this question was tangent negative or positive positive what quadrant is tangent positive in and give me the numbers because you need to learn the numbers anyways sorry I disagree sorry I partly agree yes what ah here and here because there's gonna be two answers now I'm going to find the reference angle the reference angle is going to be the inverse tangent of 1 over point 5 now familiar right same old same old reference angle 1 over point 5 63 degrees to say yes that's in your whole number which means that this angle and this angle are 63 degrees a1 equals a2 equals ready Holly here we go here we go here we go here we go here we go Holly how far oh that was easy how will I calculate that one and I'm gonna argue I think you can do that in your head you can you can use a calculator if you need but yeah they could be done 180 plus 63 to 43 yes now the reason I'm encouraging stuff in your head this first trig test is going to be non-calc right I'm Mr. Duke how will we do that step we're using a I will be eventually showing you a way that we can do certain angles without a calculator and even going backwards so but so I am encouraging you to be able to I think it's fair game to expect you to go 180 plus 63 degrees without a calculator successful if you can't suck it up raggie what trig function they give me and be really not cozy can't what did they really give me I'm gonna replace cozy can't with sign and flip the other side and now it's a question that we've done before cast rule by the way the most common mistake Roxanne kids remember to flip this side they forget to flip this side and now you've completely changed the question now you're saying sign a equals that no no no no it was coasts you can't equal flip both sides raggie what quadrants are we in according to this question a sign negative or positive what quadrants are we in right right right yeah good take it the rest of the way folks I'll do it quietly here remember don't include the negative in your reference angle I even checked my answer the cozy can't one divided by sign of 340 degrees to point negative 2.92 pretty close to that I figured we'd be a bit off because it's not exactly 20 degrees it's almost 20 and a half degrees that's about as big an error as we'll get by the way what if you put the negative in there you get weird stuff for example if you went inverse sign of 1 over negative 2.86 you get a negative reference angle which this time matches that decimal but it won't always and reference angles aren't supposed to be negative we use the negative here to figure out what quadrant we're in we don't use it here to figure out the reference angle time for the curve balls where does this method no longer work this will work for any angle on the planet except when this number here is a zero a plus or minus one or an undefined then the bets are off usually Ashley what that means is your answer is either this or this or this or this you're right on one of the four axes and those don't really have proper reference angles like the question you asked me earlier today the system breakdown example for this Andrew can you read for a to me please yeah we're gonna say theta from now on right you see that cozy can't theta is undefined this is one of the curve balls or if it had said cozy can't theta is zero be a curveball or if it said cozy can't theta is one or negative one okay our cast rule no longer is going to work instead we're going to have to fall back on x and y and r and so I say to you cozy can't in terms of x and y and r is one over one cozy can't is whatever what sorry I heard here Amanda what this question is really saying is r over y is undefined that fraction is undefined what makes a fraction undefined divide by zero if they're telling me that cozy can't was undefined what are three dots me therefore they're telling me that that was true so we put that little chain of reasoning together we're still gonna sketch but I'm gonna draw a little dotted line circle that's the arm swinging its way around what angle or angles as you're swinging around we'll have a y value of zero now your y value is how high you are what you're really saying is as my arm is swinging around when is my arm zero high here which is what how many degrees here and you can't find that with reference angles and the cast rule because there are no real good reference angles there oh and besides how the heck would you type in undefined into your calculator to find the reference angle in the first place doesn't really work cos theta is zero cos theta equals what well alarm bells gonna go off now in this unit when you have a trig equation and the answer works out to a zero or a plus or minus one or an undefined Shannon your little heart should start racing and you should be saying to yourself oh mr. duick says cast rule is not gonna work I've got to go yr and x and so Shannon now that you've calmed down I say to you cosine is what over what in terms of x and y and r it is and I want that with more authority and confidence in your voice next time what this question is saying Shannon is x over r equals zero it's saying this fraction worked out to zero what would make that fraction work out to zero it's when the top is zero the fraction equals zero the top is zero the fraction is undefined the bottom zero hey what's on the top here ah therefore x equals zero once again we can do a little sketch we can imagine mr. duick's arm swinging around in a circle in the answers to this question are where my arm has an x coordinate of zero here yes here no ah here what two angles 90 and 270 see hey what's weird about c squared now first of all put your pencils down look up I have to show you some notation if I write this I don't know whether I'm supposed to go ten squared sign of a hundred or sign of ten get a decimal and square that so as mathematicians a long time ago we said if you mean the sign of ten times the sign of ten put the squared there and I'll know you're telling me to square the actual sign decimal if you mean the sign of ten times ten put the squared there okay problem if I try typing that on my graphing calculator I get an error the only way to type it on your graphing calculator is to type this left hand version is to actually go oh if they want me to go sign squared I've got to go sign ten sign ten because multiplying it by itself that is squared sorry so when this says sign squared this is the same as an equation like this a squared equals one half what kind of an equation is this and how do I know quadratic how do I know it's got a squared now normally I'd make a quadratic equal to zero although no no we didn't do that for every single quadratic if we just had a squared term we took a shortcut we got rid of the squared how square root both sides what's the square root what's the square root of a squared what's the square root of sign squared what's the square root of one half it's not this I have to remember something from now on whenever I square root both sides what did I have to remember I heard it because when you square root both sides here the answer was actually either positive one half square root of one half or it could have been negative one square root of one half we didn't know who's in physics with me in physics we square both we square root all the time we ignore this but that's because almost always it's a t-squared can I have a negative time well I started to shut up can I have a negative time no so we just we ignore that although in the vf squared equals vi squared plus 2 ad there was at least once this year where we square rooted vf and we said oh it's going down we want the negative answer not the positive answer we actually did want that one and we rejected the positive one so in physics we're in context there here because I don't know where these numbers ever come from I have to assume plus or minus step which trig function did they give me sign and according to this question sign can be positive sign can also be what how many answers am I looking for four actually that kind of makes sense because a normal quadratic gave me two answers a trig quadratic which already gives me two answers a trig quadratic should give me four answers actually that that kind of there's some nice symmetry there because a normal trig was already given me two yes so I think a quadratic trig should give me four reference angle the inverse sign of the square root of a half or point five and I think this one actually works out bang even I have memorized a few angles I think this one's 45 yeah so what we were saying is this angle is 45 this angle is 45 this angle is 45 and this angle is 45 ready Sabrina there's my first angle how big is theta one 45 nope there's my second angle how big is theta two 135 yes yes yes how big is theta three how will I calculate it 180 plus 45 which everyone knows is 225 and Sabrina theta four this one I will make you actually figure out yourself I think it's 360 minus 45 which is 315 absolutely quadratic tricks now next unit we're actually going to spend like four days looking at quadratic trig equations but the yucky ones that you have to factor or quadratic formula here we're just going to give you a quadratic that only has a squared term it doesn't have a non-squared term so you don't need to make it equal to the square root of those that's application of number one of the cast rule and reference angles we can solve trick equations yeah Carson said I think I noticed a pattern I notice I add 90 to each one to get the next one it will always be that if it's a quadratic sine equation with a one half as your answer it's the fact that our reference angle is 45 degrees otherwise we can't use that trick every time I wouldn't bother memorizing let the obscure ones only gonna come up once in a while to turn the page three more what up Katie not only can we use reference angles in the cast rule to solve trig equations with I would say a fair bit of writing but not that much having to think it's a nice little algorithm just be careful Holly follow all the steps and it seems to take care of itself we can also use it to find exact values like example five says given that the point negative three comma two lies on the terminal arm of theta on the values of the primary reciprocal trig functions in other words they want me to find sine theta cos theta tan theta those are the primary trig functions oh they also want the reciprocal trig functions which one goes with sine cosy can't theta which one goes with cosy can't theta oh yeah and cotangent goes with tangent all right I wrote that down it's no longer blank Matthias I feel better even though I haven't really started the question at least I've yeah I've written something down what did they give me in this question Matthias what did they tell me a point what is the point that they gave me no no what's the point that they gave me what's the point that they gave me what comma what x is negative three y is two they gave me this point right here and they told me the arm goes through that point yes I'm gonna say that same thing differently what I said to you out loud was x is negative three y is two and it goes through there I'm gonna say it this way by writing it but instead of is I'm gonna use an equal sign Matthias what's x what's why it'd be wonderful if I could also use that to find our it would be neat if there was some kind of an equation that related x and y and are together fact I think this Amanda are is the square root of x squared plus y squared which is gonna be the square root of 9 plus 4 you're really reaching for a calculator what is 9 plus 4 and they did say exact values which means they don't want a decimal they want that root 13 by the way Andrew technically our should be plus or minus the square root of that but it's a radius isn't the length by definition always positive so once again we've taken this number in context and ignore the negative that is implied all right do I know x do I know why check do I know are check well then sign is what over what more specific in this case sign is what over what and cosecant will be root 13 over 2 cosine is what over what x over are in this case negative 3 over root 13 and secant will be root 13 over 3 don't put the negative on the bottom a negative in a fraction belongs on the top or in front it's just good math matters tangent is y over x 2 over 3 negative cotangent is negative 3 over 2 negative 3 over 2 by the way what's positive in this quadrant sign and cosecant are they everything else should be good is it oh I got a little error check to make sure I haven't made a mistake kind of built in skip 6 example 7 I like example 7 I like example 7 number 7 is a nice question if the tangent of theta is that and secant is positive find Jessica what are they asking me to find and I'm pretty sure it's going to be a why over our question what did they give me tangent by the way tangent is what over what what they really told me was this except Andrew one of these must have been negative which one let's find out shall we tangent according to this question Andrew negative or positive which means I'm either here or here and secant according to this question what positive I don't have a rule for secant what a secant go with cosine this is really saying that cose is positive so where is tangent negative and cosine positive well cose is positive here and here where's the overlap Holly if I'm down here is my x positive or is my y positive where did this negative come from I'll give you a hint listen to what I say if I'm down here why is negative isn't it that negative this I didn't know who it belonged to came from there which means sine is going to be negative 3 over oh do I know x do I know why Amanda is there some lovely equation that relates x and y and r together really what or as I like to say r is just the square root of x squared plus y squared which is the square root of 2 root 5 squared plus negative 3 square and it relax let's do the easy part first Steph what's negative 3 squared now what's really bugging you I'm guessing is this this is actually nice and easy to do you know how I do it Steph what's 2 squared what's root 5 squared root 5 times root 5 is just what not 20 it'd be root 25 which is what 5 that did it get ready see what I want to take you what's 2 squared say that's allowed what's root 5 squared just plain old it's a 4 times of 5 because it's 2 times root 5 all squared is going to be 2 squared times root 5 squared it's 4 times 5 in fact in my head no calculator required what is r what is 20 plus 9 and r is root 29 think about what we've just done we don't know the angle yet we were able to find and they could have asked us any trig function we could have found in fact we did find any trig function one question ago planning all the trig values as exact values even though we don't know the angle we need to practice this I'm fortunate to give you lots of practice on this one all two all three all four all five all duck yes six all the duck seven all I'm gonna skip eight oh mr. you are so kind nine so one to seven nine you have the remainder of class my children hopefully that made sense to you though