How to solve trig equations with a calculator and the unit circle

in this video I'm going to show you how to solve equations like this one and the cousin tan as well and four degrees and also radians I'm going to show you how to do it if they ask you to solve for theta to the nearest minute and I'm going to be showing you how to use this cashier and ulcer I've got an HP and TI so I'll show you how to do all these things with several different models of calculator now the first thing you want to do when you're solving trig equations of the calculator is make sure that you're in the right mode so we're looking for an answer in degrees now the way that you tell if your calculators in degrees is for this model there will be little there when it's in degrees right now at the moment I've got an AA little capital that means it's in radians mode so I'm gonna have to switch it into degrees so the way you do that on this model is you want to go to setup so I'll hit shift set up and then you can see degrees is number three so we hit three and now have little diva sorry now it's in degrees under the HP it's kind of similar where there's an old there if it's in degrees word if you've got an there instead then you'll need to go out to the mode hit mode again and then hit one four degrees and now you're going to do that for the ti it actually says deg there so that's in degrees if it didn't say to EG if it said rad for radians you would go to mode and then you can use the arrow keys to select so if hit enter yeah that's going to search for three Nan's got such back to degrees and now it's in degrees once I've done not what would do is draw the unit circle so we can get an estimate for the solution and see what quadrant they're in because there's lot of things that can go wrong when you're doing these on calculator could pop them that being in the role murdered you might just type something in wrong and it's really good to have that estimate so that you know if you've done something wrong so if you haven't seen my previous video where covered solving trig equations with exact solutions you might want to watch that first but for this one you've got sine theta equals point 2 so sine is the y-coordinate so we go up to roughly 0.2 which is right now I'm just drawing horizontal line like that and then where that cuts the unit circle is the two solutions that were looking for so mean that looks like maybe 1020 degrees so we could estimate theta might be it's like 10 degrees or might be 180 minus 10 just 170 so it's good to kind of have that estimate in the back of your mind so that you know if you've done it wrong but what we can do to work out where it is exactly is you need to use this sign to minus 1 let me just explain what that means so if you've done function notation before you might be already familiar with that like if you do the minus 1 of you might have seen that before know what that means otherwise just explain why they use that notation so if you had something like some variable times theta equals 0.2 you could solve that by dividing both sides by and now dividing is the same thing as multiplying by the reciprocal of which you can write as to the minus 1 so it'd be 2 minus 1 times the 0.2 so when we solve an equation like this it's got the sine theta equals point to sort of with an analogy with this we say that theta equals sine to the minus 1 of pointer now just want to point out that this isn't literally the reciprocal of sine because that doesn't even make any sense like the sign here isn't variable it isn't number it's actually function it's sine of theta it's not sine there but they use this notation because it's kind of analogous to this but it doesn't actually mean whatever sign it's sometimes called oxide so you you might sometimes see it written as oxide of 0.2 or it's also sometimes called inverse sine now the way to solve that on the calculator is to get sine of minus 1 we got shift sign and then type in this appointment sure and then that tells us all of 0.5 now it could already tell you one of the solutions it doesn't know which one you want so that's just gonna be this solution now we need to run that to the nearest degrees that would be 12 degrees and then to work out the other one it's just 180 minus 12 so that's type that and you can just go 180 ways that's what - the last answer so 168 now it's more or less the same on the tih peso I'm not going to show you those as well all right let's have go at cos theta equals negative 0.6 now if you want to give this go for yourself then pause video yeah all right first thing would do is draw neon suckle causes the quarter and it's negative which means we go left five point six around now I'm sure that in and then we can see these are the angles that I'm looking for so on the calculator we can go in that's close negative zero point six and that gives us hundred twenty seven if you round that off to the nearest degree you now to work out the other one if they said negative 180 to 180 you would just say negative 127 but in this case to get it into that range what you can do is add 360 to negative four and 27 so the easiest way to do it from here is just go to 360 - ants I'll give you two threes and that's it alright let's try tan theta equals negative two so what would do is join the unit circle now for tan what you want to do is foreign it's tangent line and then this is the coordinate on that night so you go down to negative two which is found now and then you want to draw line through this point and through the origin just said that's right through and then these are the two angles that I'm looking for so this one looks like maybe on June 28 degrees ish and this one is maybe like negative 60 which would be 300 in that range so that's again really useful to have those estimates so that you know if you've made mistake on the calculator so to work them out exactly we won in this town of negative two and that tells us the hang of 63 so the calculator doesn't know what range you and the answer in so to get it into that range we would have to add 360 and that tells us when there's degree that would be 297 degrees so that's that one now work out the other one think of it like that is 180 degrees so if we just subtract 180 from that that'll give us the second outside so let's go - 180 and that tells us 178 degrees so that's how you sort that one all right same thing that equals negative still point nine this woman may be tad trickier than the other ones so don't want to deprive you of the challenge of working this one out so pause the video here and see if you can figure out the solutions to that now first thing would do is draw the unit circle and we've got y-coordinate of negative point nines let's go down here and then these two angles that we're looking for so we might expect it looks like about 70 degrees maybe so we're expecting it to be like maybe hundred eighty percent so 250 degrees ish might be where the solutions and then the other one is like 270 plus 20 to 90 so it's really good to have an idea of what we're looking for first and now let's try oxide or negative zero point right next negative 64 now that's not in the range of one sir then we could add 360 so 296 it's one of the solutions now to get the other one there's actually lot of different ways you can do this I'll just show you couple ways maybe what we can do is go back to what we had before take that 360 off again so negative 64 now if you think of that as like continuity plus the 64 that's like hundred eighty minus negative 64 so you can vary like 180 minus cents and that will give you the other one 244 degrees another way to do it which might be bit easier actually is not to work out that angle directly but actually to work out what this angle is as positive angle and the way you can do that is very back sine or positive 0.9 and that tells you what's called the principal angle which is like that angle and that angle they're both 64 and then you can go you know 180 plus that it's gonna be that one and let's just get ready 180 again and then 360 minus the 64 it's gonna give you that one so 316 and is that way yeah 296 so that's couple ways to do it you might also have done something completely different you might have done something like arc sine of 0.9 and then maybe did 360 plus that and then maybe you thought well to like flip it over to there we do 180 minus not right sir let's try 180 minus that and then no that's not in the right range let's add 360 to it so you've got all sorts of different ways that you can do it now don't want you to feel like you have to like memorize one procedure instead of lock into that because you know the more you memorize the less you understand and think think it's better to just stay flexible and you know just use your creativity to come up with the solutions you'll know if you've done it right if it if it looks like your estimates that you did before now let's see what happens if they ask you to solve it to the nearest minute so first just want to give you some historical context for that term minute because he might be confused like isn't there measure of time nor angle well when they were first dividing the hour into little parts they called the first the first division past minute prima in Latin which means the first small part and that was sixtieth of an hour and then they wanted something smaller than that so they called it past me address corner the second small part now it gets bit annoying saying that praise are never again so they abbreviate this down to minute and this down to second its minutes as in minut as in small so that's where that word comes from and it's actually not just the time it's also for other things like angle just wherever you need something to be divided into 60 parts you might call it minute and then if you want us something smaller where that minute is divided into 60 parts you call that second so for example if you wanted to write like 32 and half degrees you would say 32 degrees and then half degree would be 30 minutes so you write that now you write that with little what's called prime symbol there so they use prime as in like prime is in first so the symbol for that was prime and then for second is like that mean they're basically just tally marks you'll also see these in calculus when you get to calculus you'll see like prime of meaning the first derivative and that meaning the second derivative now these days we pronounce that double prime but back when they were sort of inventing this notation they would have actually said second for that because that's literally what it what it is it's like to that now to solve the equation that's draw the unit circle we're talking about cuz that's the coordinates so we go to equals point three draw that in and then these are the two solutions that we're looking for okay no let's go in the sponsor point three okay so that's if we were answering to the nearest degree that would be seventy three degrees but if we're answering to the nearest minute there's really helpful button here you can see the the symbol of minutes there so that says degrees minutes seconds so you can just hit that button and that will convert it into degrees minutes and seconds rather than as decimal so we've got 72 degrees and now we have to round this off so we have to decide if this is less or more than half minute now remember that there are 60 seconds in minute so half minute will actually be 30 seconds so this is more than half so we actually have to round that up so don't be 70 degrees 33 minutes if they asked you to do it to the nearest second you would say so it in degrees 30 minutes and we'd round that off to 33 33 seconds like that okay so that's that one now if they said negative 180 to 180 you could just call this negative 72 degrees 33 months but since we need to be in this range so we can't have negative answers we can guard on 360 - so - Eddie segment except we have to come out that into degrees nuts so we hit the the button for that and we can see that's 287 degrees and 27 minutes you're Brown that one down because that's less than 30 okay so that's how you do it on the Casio on the HP it's basically the same but on the TI it's little bit different so you're typing in up cause about 0.3 and time and then to convert that into degrees minutes and seconds you need to go to the angle thing yes so it's good angle and then if you scroll down the bottom you'll see one that says DMS now that stands for decrees minutes seconds so we want to select that one you don't actually have to scroll down each time you can just press six you can see the six there mr. MS so that connotes DMS that enter again and that will tell you so many degrees and 33 minutes if we round that off okay and then to work out the other one when we go to 360 - that's tells us that and then we have to convert that again so angle and then I'll just hit six this time and that will come about that so 287 greatest ways of us let's draw line radians sorry that's ten you'll enjoy that line in and then we'll go to y-coordinate of 0.1 around there and then connect that origin going so there's the two angles that are looking for us and maybe 10 degrees and except that we're not walking right in degrees or looking ratings so some small angle and then something bit bigger than PI so it's like 3.2 maybe now what we need to do is change this into radians goodsoon degree you know it's sir your setup and radio that's fast okay and then we can go octan ops or 0.1 so point zero nine so if we want to round that to two decimal places that means we want to like cut it off there so that gets rounded up to like 10 so it's actually 1-0 safe theta equals zero point one zero would be the fast solution and then to work out this one because this angle here is 180 degrees or PI if we're working in radians what you can do is just add PI to that answer and that will give here three point two four so that's how you answer that one let's try sine theta equals the negative point seven for theta between negative PI and PI so pause the video here and see if you can figure it out before you watch my solution now the first thing would do it gasps draw the unit circle sine is the y-coordinate so around two United points seven just there so now we know what quadrants we're looking for that's really helpful let's go to the calculator ox sign off negative four point seven I'm still in radians murmurs that's fine now that's that's alright because now we're in the range negative PI to PI otherwise you'd have to add 2 pi to it to get it into the range zero to two pi if that's what they are asking for but for this one it's fine dancer ma seven point seven eight that would be if we round it to two decimal places now to work out the other one what you could do yes go well pi minus that is sort of how think of flipping that over but the problem with that let's try that the problem with that is that's now outside of the range we want that's actually describing it that's like going around that way pi in bit more so what we'll have to do to get it in that range is subtract 2 pi from them so you might notice that you know this minus 2 pi an easier way to do that would have been negative pi minus ions which also makes sense so if you thought of that that's fantastic all right but it doesn't really matter as long as you get there in the end so negative two point three seven would be the answer to that one let's go back to degrees that's so tan theta equals three for theta between negative 180 and 180 to the nearest minute so pause video here to figure it out for yourself now what would do is draw circle for ten you want to draw in tangent line and then we want y-coordinate of three so that's one like one two three and then connect that to each other so these are the two angles that we're looking for maybe something around 70 degrees and then this would be like maybe 100 negative 110 if we want it in that range so let's try that out exactly uptime of radium they tell us that that's good and then if we learn to the nearest minute we hit what don't know we've got 71 degrees and then we have to round it up cuz that's big so 34 minutes and then if we want to work out the other one if you were doing 0 to 360 you would add 180 to get around there but since we want we want to express this as negative angle what we can do is subtract 180 so there's no need to like memorize these things you can just kind of like look at the diagram and figure it out use your creativity it's the best way to do months think and then we need to put that into minutes in seconds but may have 108 degrees and so that's less than 30 so we round that down to 26 minutes in the next video in this series I'll be covering slightly more complicated equations like this one and also general solutions to trigonometric equations so check that one out and if you enjoy this video then give it thumbs up and let me know in the comments if you still have any questions