 Why are we not surprised lesson three? What we're gonna do today is we're gonna start just barely to learn Why math nerds find trig so useful it is but I would say I was gonna say by far One of the most useful branches of math So far we've learned how to name triangles a quick review It says fill in the appropriate name in the blank if that's my angle this side right here opposite adjacent or hypotenuse I'll use a letter H This side right here if this is my angle right here opposite adjacent or hypotenuse I'll symbol that with an O this side right here opposite adjacent or hypotenuse I'll symbolize that with what do you think? No, Q. Are you yeah? Okay, and a What if instead don't write this down Devin don't write this down What if instead I was talking about this angle with the hypotenuse change? Nope, would the adjacent change? Yeah, now it's the opposite and that would be the adjacent What if I was talking about this right angle here? I never will because as it turns out It's yucky Courtney opposite and hypotenuse are the same. We can't use a 90 degree angle We also I think noticed last day that 90 degrees doesn't even work for tangent. It caused our calculator to crash That's why So it says we've learned how to find a ratio given its angle or find the angle given its ratio Use a calculator. It says to find the following sign of 35 Try it on your calculator. This is also your way of making sure you're in the correct mode And you should get point five seven three five seven six four three six four blah blah blah I'll go point five seven Right Boston you got that okay Okay, find the cosine of 15 point nine six six and then see You'll notice here. We're going backwards Sydney here. I don't know the angle. So we're going in the opposite direction How do I go backwards and find the angle when they give me the trig ratio? second function or shift or inverse whatever that one is top left usually on your calculator three point one four five Seventy-two degrees seventy-two point four. Let's just go seventy-two degrees Now let's start to apply this what we're going to apply this to then is solving triangles and Then what we're eventually going to do is take those triangles and actually say look instead of a triangle on a piece of paper Let's pull this triangle from architecture and use it to solve something in a building or in a situation very useful How do we solve for triangles Jordan? There are three steps. The first step is label the sides Opposite adjacent hypotenuse The second step is to pick the trig ratio, but we had a handy-dandy acronym to help us remember which trig ratio went with which Does anybody remember that little handy-dandy nine-letter acronym Marcus? Let's write that here. S o h c a h t o a I will not give you that on a test I will expect you in this unit to know Somehow either through memorizing or through the acronym Sierra that sign is opposite over hypotenuse see it and that cosine is what over what? Over hypotenuse and tangent is what over what? Okay Easiest way to do it whenever you turn the page in your homework right so katala at the top of the page That'll be enough times to get it in your memory and when you get the test You know what the first thing you should do is at the top of the page Right so katala We are going to learn the cosine law and the sine law which are longer and yuckier those I will give you on the test I won't make you memorize those This yeah Yeah, and then we solve to solve. We're gonna almost always end up cross multiplying or If we're finding an angle We'll use that second function or shift or inverse button It takes way longer to say than to do now in our notes We're going to show every step Joe my physics 12s in all honesty I would hope that they could actually go straight to their calculator showing no work and write down right away that x equals Something and I say in their homework. I encourage them to on a test or a quiz Do I encourage them to show more work? Yeah, because if you do make a dumbest rake you want the part marks I will say that to you guys as well, but you might take you a while to get that good All right. Shea. They gave me this angle here opposite adjacent or hypotenuse Let's put an o right there. Let's put an o right there. Shea Adjacent or hypotenuse Now I see some teachers when they teach this in grade 10. They tell you to label all three sides Don't so some teachers say Go like that. I think that's an unnecessary step. First of all, I'm not going to use it So why would I write something? I don't need that's wasted time and also to me? That's more confusing because now I think if I do that Shania, can you clearly see it's ONA? Where if I have That there you have to think oh this one's blank. I'm using these two. It's ONA. I think you just confused yourself So don't bother putting the H there. That's my preference Why is that so helpful? What letter Shania? What letter? Which trig function has the O and the a and nothing else in it? Let's write down tan What always goes next to the trig function the angle what always goes next to the trig function the angle Devon What always goes next to the trig function? Emily what always goes next to the trig function? Matt you know what always goes next to the trig function call me silly, but Jordan, you know what always goes next to the trig function What's the angle? Yep, Shania, what does the O and the a stand for what's the opposite side ah Over what's the adjacent side and we did these last class we said to solve these It's cross multiplying Can I stop writing the over one can we please just remember that it's there and that it is a fraction Is that okay? You can write the over one if you want to but I'm probably sometime in the next couple of days Gonna freak out and stop doing it. Is that a hand up? sorry over 18 Because the angle is 18 and I did a sloppy mistake didn't I? Jordan you get a candy later Now I cross multiply Cross multiply means go that times that equals that times that this is the easy one. What's one times x? Hmm x Equals what's 15 times tan 18 why that's 15 times tan 18? I just write it out is the x by itself Then we go straight to our calculator What's x? How big Would you get Jordan 4.87 anybody else? No, yes. No. Yeah 4.87 Now if you're not getting that Your calculator might be in the wrong mode. So if you went 15 Times tan 18 and you didn't get 4.87 now is your time to raise your hand and let me know And I'm looking in the back corner. Is it working? Okay, or not? I have no idea what you're saying. Sorry. I Think you're in the wrong mode to me pause for one second 4.87 solve for the missing side or angle. Okay This time Danielle the angle is up here The x opposite adjacent or hypotenuse I totally agree. Let's put an H there What about this three opposite or adjacent? Which one and convince me Opposite means facing opposite would be that guy Right right now you and I are opposite each other We're facing each other Adjacent means beside you and Matt are adjacent to each other Which by the way Danielle is how I remember it by the way Matt. I know that's the opposite side I'm not gonna label it. You know why cuz A and H makes it easier Danielle Which of these has the A and the H and nothing else in it? Cosine did you say cosine? You're right. Oh Let's write that down cosine Danielle what always goes next to the trig function What's the angle here good? equals and What is the ah? Stand for more specific in this triangle here three over what Yep, okay. I can see a few of you want me to put the over one. I will one more time I'll stop eventually here How do I solve this cross multiply? how What's three times one that I'm expecting you to do in your head Alex What's three times one and x times cos 56? I just write as x cos 56 Do a little better x than that Now here when I asked is the x by itself Jordan said yep Here when I ask is the x by itself is Jordan gonna say yep. No Jordan what else do I have on this side? Costs is not what we call it cost is like when you're sneezing or coughing cost cost Cosine or coasts. I'll take co-cosine What's happening between the x and right there adding subtracting multiplying or dividing if I don't write anything what did that always mean? So how will I move it over? Turns out x is gonna be three over cost of 56 Emily three divided by 56 cosine button equals Right try it please could be you need to learn how to use your calculator What do you get three divided by cosine the 56? You get five point three six four eight seven four nine five blah blah it goes on forever without repeating it Yes, so five point three six. I said if you're not sure two decimal places, it's pretty good Turn the page or next page over Okay This time they want me to find an angle You know how I know because there's an angle, but it's not a number. It's a variable. It's a Greek letter Alpha can you see our lowercase a comes from it in fact This is a Greek letter beta What letter comes from beta? Do you think and the word alphabet literally means in Greek a b Cs we say ABCs in English, but the word alphabet comes from the alpha beta. I Didn't know that yeah, and you learn something new your teenagers. You don't know much Finder here's my angle right here Opposite adjacent or hypotenuse. I totally agree adjacent or hypotenuse I Totally agree which trig I know I could label this one not gonna. It's blank. Which trig function has the o and the h in it Thank you for saying sign and not sin. It's a sin to say sin when you may not mean sign Binder what always goes next to the trig function the angle. What's the angle this time? That weird symbol. That's why I always say what always goes next to the trig function say the angle in this case the variable equals What over what so here's the question? How do I find the angle? This is not gonna be cross multiplying Now Binder said you divide it and then go shift. You know what we're in grade 11 Most you have good calculators. How many of you have a calculator that for this question here? You actually type 15 times tan 18 equals in that order Okay, so those that have the better calculators you can go like this Second function or shift sign bracket Open bracket 11 over 12 close bracket and that way we can do it all in one step and am I lazy enough to want to do it all in one Step nod your head nod your head nod your head. I am Okay, those you that have the cheaper calculators you're gonna have to go like this 11 divided by 12 equals and then shift sign That does that work? I hope you should get 66 point four four nine three five Yes Yeah, okay You're not the only one by the way. You're just the funnest one to pick up so I get Alpha equals 66 degrees usually if it's an angle if they don't say I go to the nearest degree because one degree is really really really small One degree is only about that much like point one degree. I don't even know if we can see it within a good Now it says all right Let's try it with some you know what I'm gonna cross out the word easy because word problems in general are difficult Let's try some applications When doing word problems remember What adults stand for draw the little picture first draw the picture then label ratio solve We're going to use two pieces of terminology Often we're going to talk about the angle of elevation. That's measured from the ground up That way and sometimes we'll talk about the angle of depression The angle of depression is measured From your eyes Downwards this way and the way I remember that she is I always think angle of Depression and when I'm depressed don't I look like this? My eyes go down right you said Elevation I can remember because elevator goes up so if a question ever talks about angle of elevation ground up angle of depression From wherever the person is standing downwards Example three five sorry an observer standing 3.5 meters away from a flagpole measures an angle of elevation of sixty to three degrees to the top of the pole find the height Dump What does Dump stand for Sydney? You know it's gonna appear in my trick picture What shape do you think is going to appear in almost every one of our pictures in the trigonometry section a? Triangle captain obvious. Yeah, so you know what I'm gonna go like this yank. I think this is my flagpole You can get fancier. You can put a little flag on there if you want to If you're artistic go ahead. I just rec diagrams if I try and go artistic Devon what are they asking me to find sorry Devon to the left. What are they asking me to find here find what? Okay, you said the height of the flagpole is that this side is that this side is that this side? Did they give me an angle anywhere and angles are pretty easy to spot because they have a degree symbol next to them Devon Is there it yet? What? Now what kind of an angle is it angle of elevation or angle of depression? That's from the ground up and I guess the person is standing like right here Here's the person This 3.5 meters away from the flagpole Here or here Slanty or the bottom bottom have you guys turned the page yet, or do you still have so katoa written on the previous page? Do you okay? Otherwise I would have said let's write it at the top of the page. We got so katoa there Let's label our triangle Marcus opposite adjacent to my pot news. I Agree how about here which trig function what I was doing at Newport Beach on Sunday Yes, what always goes next to the trig function? What's the angle here? Equals what over what Marcus? Good solve this on your own. I'll solve it up here. Let's see if we end up at the same spot. I got that I got that I'm frozen now. Whoo. Thank you. Am I right? I think yeah, two more were done by the way I believe if I recall my days of the year. I believe yesterday was international kite flying day Everyone was supposed to go fly a kite yesterday. So this question is very timely a kite is flown with 210 meters of string The angle of elevation of the kite is 61 degrees. How high is the kite? Joe, you know, I'm gonna do first dull What's that stand for? Okay There's my kite Boston. What's this question asking me to find? I think it's reasonable to say that yeah How high the height and I think we have probably Someone standing here Kind of like that. Is that okay? Did they give me an angle? Yeah, Joe. How big in the question. Did they give me an angle? Sorry Now is that an angle of elevation or angle of depression angle of elevation or angle of depression? So that's from the ground up. I think that means they're telling me put it there Erin can you read to me the first sentence for me kiddo? How much string? Where is the string of the kite this bottom one or this slanty one? I'm assuming you've all either flown or seen a kite be flown Bottom or slanty. How does the string of a kite go slanty? You know what? I think they want me to put the 210 right there You see how I was able to kind of reason my way there Is that okay Boston? Okay Erin let's keep going here a little bit this X opposite adjacent or hypotenuse. I totally agree This 210 adjacent or hypotenuse. I Totally agree. I'm gonna leave this bottom one blank. I don't want anyways Erin joy of my heart light of my life which trig function as an O and an H in it sign. Let's write that down What always goes next to the trig function? Equals what over what? Yep. Yep. Good find X you got a hundred eighty three point six seven It's a high kite. It's up there Almost two whole football fields Yeah, nothing. Sorry Sydney Wait for failure to get your adrenaline rushing about 30 seconds or so. Oh, that was a great look. He just gave Looks could kill a via corpse last one Example seven. Mr. Dewick is standing on an observation tower 30 meters above the ground. He's finally snapped He sees Ryan standing 120 meters away from the base of the tower Find the angle of depression. Mr. Dewick has mr. Ravel forgot to change that name. Mr. Dewick has to aim his nerf trademark Dark gun in order to hit Ryan. So So I'm standing on my observation tower and I've had enough I'm gonna finally snap and shoot someone named Ryan with my nerf dark gun. I Think about it. They want me to find the angle of depression. Look look look look look look look look look Do you see the angle of depression? I think this one is going to be from the top down That's why we call it the angle of depression Okay Simon, what are we gonna do first? Dolp you back with me now? Because I don't know where you were What does Dolp stand for Simon my friend? What is Dolp stand for Sierra? Let's draw a little picture There's the tower Yeah, how high did they tell me? 30 There's poor Ryan There's my triangle So we put the 30 on here. I Also see 120 says 120 meters away from the base of the tower Sam is it's 120 gonna go on the slanty part or the bottom part Sorry, I totally agree bottom Sam, what do they want me to find? Look up folks. They want me to find this angle here I Traditionally Courtney use Theta zero wearing a belt as my variable of choice for an angle But Courtney we have a problem. I have a triangle down here. My angle is up here Joe are you ready? Look up look up look up What letter? That angle will be the same and that's in my triangle. So even though they want me to find this Matt I'm gonna use alternate interior. I'm gonna find this all the geometry stuff is handy Yeah, it is actually that's the angle. I'm gonna use Taylor my child if that's my angle opposite adjacent to hypotenuse How about here which trig function what I was doing on Sunday in the beach. Yes Taylor what always goes next to the trig function? The angle what's the angle here? It's a variable theta Equals what over what? 30 over 120 by the way Can you guys see why I had to move my angle down to here because this angle here doesn't have any triangle going with it I do a tiny bit of geometry Okay Am I cross multiplying here? No, I'm finding an angle. How do I do that? And you may have to use brackets Mr. Dewick. Yes, Emily How do we show in our work that we've hit the shift button? Emily look up When I press the shift button My calculator puts a little negative one right after tan kind of like an exponent That's the symbol for hitting the shift button. I'm going inverse tangent of 30 over 20 inverse tangent of 30 over 20 And I get 14 degrees. I Get 14 degrees so Simon 14 degrees Of course, I'll add a little over Dad 14 degrees Wait good What's your homework? Hey, did you all just turn the page? What should you all right now be writing at the top of the page if you're in any way clever? Skip one number two Three a 3b 3c is good 3d is good 3f is good. I skipped e G is good. I skipped h J is good Okay You got lots of time finish the homework in class You