Algebra 2 - 9.1 Intro to Trigonometry

0:04 All right, here we go. Let's start our next chapter here. We're going to talk about some trigonometry. Love trigonometry. This is pretty awesome. Also love Austin Powers. Got a little picture of uh Dr. Evil here and Mini Me. Maybe thinking, "Hey, what's that all about?" Well, I'm going to do a quick review of trigonometry stuff you've seen before hopefully and ring some bells. And it all starts off of similar triangles. And what are similar triangles? It's just like mini me and Dr. Evil. Same shape, different size. So they look the same, different sizes. So let's get going and look at some of these triangles. So I've got two similar triangles here. They are similar. They're the same shape, different size. How do I know the same shape? Well, they've got this right angle down here and a 30° angle. And just like this one has a right angle and a 30°ree angle. So they got to be the same shape. So the same triangle, this one's just bigger.

0:49 Awesome. So what we used to do or with similar triangles, we say, "Oh yeah, I can say something like this." is like, oh, I know that, hey, this long side here matches this long side here. So, let's say I want to find this question mark. I don't know this side. So, I can say something like, hey, 60 is to 70 as what? What are the corresponding signs here? Short side over here, little guy is to my question mark. I'm going to call my question mark x. So then I can solve this. If I know the ratio, they're proportional. So, I can solve this. I can say, "Oh, yeah, that's just 60x = 2100, 30 * uh 70 and 60 * x." And then if you go to your calculator, type that in real quick. Clear out any of that stuff I got in there. Clear maybe. Clear. Clear. There it is. A little slow moving there. So, if I have 2100 and I

1:36 want to divide it by 60, and some of you may have already guessed this, the answer is 35. And maybe you saw it already. And how did you see it? Well, you use this ratio like, yeah, 30 is half as big as 60. 35 is half as big as 70. But the idea is any triangle that is 30 and a 90° angle is going to have the same ratio. They're all going to be proportional. So, how does trig come into play? Like, that's great. How does trick come into play? Trig comes into play was like, hey, what if I didn't know that? What if I only have one triangle here? So, what we did is we went and just found all these ratios. What is this ratio here? So, again, let's pretend like I didn't know it was 35. Here's what's great about trig. I don't need two triangles. I can say, "All right, if this is my angle here, 30, and I know, and I'm looking for the one that's opposite of it, this is the

2:22 opposite side, and I know the hypotenuse. Well, what trig function is that?" And I'm going to review all this, is sign. That is the sign. So, the sign of 30° is what? Let's pretend I didn't know, even though I know the answer is right there. It's x over 70. So, what we do is the calculator stores all these values. We figured them all out. calculator stores them all. So if you want to know, hey, what is the sign of 30? Well, the sign of 30 is going to tell you that ratio. This is what a trig function really is. So you probably done these, but maybe you didn't think too much about it. That ratio is 0.5. So that's what's super cool here is the sign of 30 is 0.5 and that equals x over 70. And if I want to solve this, what do I do? Times both sides by 70. And what do you get? Half of 70 is that 35. So this

3:08 works for not only 30, we can type in the sign of 32, 37, any angle we want. We can find the sign value of it. So we don't need the other thing. I can see yeah sure that ratio makes sense. That's what I'm keeps. I must have said ratio 40 times. The ratio of these two things. 30 over 60 is the same thing as 35 over 70. That's what a trig function does. That's why a trig's so awesome. It's like similar triangles, but you don't need your mini me now. You can just do it with Dr. Evil, if that makes any sense. All right. All right. So, here comes your review. So, you've done this in geometry. If you want to go back and review, I'm going to go through it quickly, just assuming you've seen it. If you've never seen this stuff and you want a full on refresher, go back to our geometry course. I think it's chapter seven in there in the traditional geometry. Check it out. It's going to go way more in depth. But here's what you

3:53 need to know. We need to know uh Sakatoa or Sookatoa. I don't know how you pronounce this. As long as you can remember it and say it. Uh, a lot of people remember the chief sakatoa or some old hag came around here taking our apples. However you remember this, it's sign is opposite over hypotenuse. Cosine's adjacent over hypotenuse. Tangent is opposite over adjacent. So, we're going to use this thing called theta. This little uh weird ang this stands for an angle. It just is pronounced theta. So, if you didn't use theta in the past, we're going to use theta to represent angles. We could also use x or all kinds of things for it, but this is the angle. So if I'm referring to this angle and I have a triangle with sides like A, B, and C, the easiest side to find is always opposite of the right angle. Here's your right angle. So the one opposite is your hypotenuse. So I always find him first. Now, if I'm

4:40 talking about this angle, then it's easiest to say the one straight across is your opposite. Now, if I was talking about this angle up here, the opposite would be over here. But I'm talking about theta. So that's the opposite. And then the last one, whoever's left over is the adjacent. So if I'm looking for the sign of theta, I'm looking for the opposite over the hypotenuse. This sign is opposite over hypotenuse. So in this case it would be b over c. So whatever b is you plug it in whatever c is plug it in. How about the cosine? If I'm trying to do cosine in this case it would be a over c. The adjacent over the hypotenuse and toe would be opposite over adjacent. That would be b over a. So that's like the generic formula way to write it. Awesome. One little sidebar over here. I know it says sin, cost, and tan. We don't say sin. That's weird. Uh or cost.

5:27 We definitely don't say cost. It's sign. So, just know that these are short. We just are abbreviated. I know like sign's not a long word. We're just dropping the e. But please, you got to say sign, cosine, and tangent. It is not sin and cos tan. We let that one slide. He's lucky because it is kind of short for tangent. So, let's use these in real triangles. Sometimes I'm say, hey, can you find the missing side? So, again, I'm doing this quick. If you need more, go back and uh and look it up. Uh so I'm talking about this angle here. This only works for right angles. This is the only when s cosine tangent works. So I got to figure this out. So if that's my right angle, even though I spun it around, boom, this has got to be the hypotenuse. Then if I look at this over here across, this is my opposite. And then the leftover guy's got to be the adjacent. It's the one touching him. So if I have

6:14 this, what do I know? I'm looking for x. I know this. So I'm looking for the hypotenuse. I know the adjacent. So who is adjacent and hypotenuse? That is cosine. That is this one. It is adjacent and hypotenuse. So I'm going to say, hey, the cosine of 28°. So it's the cosine of this angle is what? It is the adjacent angle over over x. Awesome. So to solve this, don't freak out. It is just this cosine of 28 is just a decimal. So we can put it in right now if you want. You're like, hey, what is the cosine 28? Let me have that ratio. The problem with doing it now is you're going to have to do some kind of rounding. Look how big that decimal is. So if we don't if we can save it for later and not have to write that all down, our answers will be more precise.

6:59 If you want to write down now, you could. So what we normally do is we leave it in this form here. Cosine 28. I'm solving for x. I hate x in the bottom of a fraction. So what do I do? Get rid of it. You got to times both sides by x. So that's good for get getting rid of the fraction. So I'm left with 22 over here equals. Again, this is just some decimal. This is just 88 something blah blah blah. But this is the precise way to write it. So I've got this. So how do I get x by itself? One last step. I'm going to divide both sides by that ratio by that decimal. And I'm going to do something like this. Cosine 28. Now those cancel and I get x= 22 / cosine 28. Now I can type that in. So if I wait I'll say 22 / the cosine of

7:44 28 and you get something like this 24.91. And then I always say hey does that make sense? Can the side really be uh the hypotenuse be 24.91? Well just remember hypotenuse is the longest side so it better be your longest side. Awesome. There's probably one or two people out there wondering like man I got a totally different answer. Let me just show you this real quick. If you got a crazy answer for the cosine 28 was not this or if you got a weird answer, you have to check your mode. So if you go to mode here, make sure you're in degrees. We're going to be talking about degrees this whole time. If you happen to be in radians, you're going to get a whole different thing. It's a whole different measuring system. So just double check your calculators in degrees, so you're with me and you're not getting weird answers. You definitely would have got 24.9. So that's why I was kind of double check make sure we're cool. That's

8:31 coming up though. Radians. Love it. Love it. All right, let's go back. So, we found a missing side. What about this? Can we find a missing angle? Sure. Same thing. I've got to kind of relate these uh these sides here. So, this is the angle I'm talking about now. And I used x instead of theta. Totally cool. And if I look at this, this is what the opposite. And I know that guy's got to be the hypotenuse. So, who is opposite hypotenuse? That is the first one. That is sign. So I'm going to say, hey, the sign of some angle x or theta, whatever you want to do here, is opposite 12 over 20. So again, now I'm saying, hey, uh, now I know this decimal. Now I know the ratio is 12 over 20. If you want to make that something a decimal, you can. Now 12 over 20, it's going to be 6. Uh,

9:18 especially if you just reduce the fraction. But it's just this decimal here. It's just this. So I want to know what angle makes this. So your calculator will tell you. We didn't have to do any of that. We could have just said, "Hey, instead of saying the sign of some angle, I want to know what angle makes that sign." So, hit second sign. It's the inverse. We're going to say the inverse. We're going to do it the other way. Of 6 or even just I usually just type my fraction in there, but you can put your 6 in there. And this should tell me if I'm in degree mode. Yes, 36.86. We're just going to kind of go two decimals and quit writing. So, 36.86 degrees. So, we're going to truncate. Just cut it off after two. That's what x is. And does that make sense? Is that angle? Yeah, seems reasonable as a 36.86 uh degree angle. Awesome. That's it,

10:04 man. One's find a side, one's finding angle. What if I kind of put it all together and I say, "Hey, can you solve the entire Whoa, what is that?" Uh, Dr. Evil, also Dr. Sullivan. And pretty uncanny uh resemblance there, isn't it? That's kind of weird. I think he has a has a future in acting there. Uh, let's check it out. If I want to solve this triangle, it means find everything. So there's lots of different ways to do it. Uh but I want to find all the angles, all the sides. So what I would do personally in this case is say, hey, if I know this is 62 degrees and this is 90, triangles always add up to what? 180. So I know that 62° plus 90° plus h let's just call this x is got to equal 180°. So that's one way to do it. We're

10:51 looking at 152°= x. And then just solve this bad boy. If you subtract that, you get what? 8 to 60. So 28 degrees. So without really doing much work at all, I didn't have to do any trig functions. I knew this was 28°. Awesome. Super cool. Now I'm here. Now I can find these other two sides. So let's just use the 62. Either one you want. If I'm talking about the 62, well 12 is always hypotenuse is across from the 90. So if I want if I do sign, what am I finding? The sign of 62 is what? Well, in this case, if this is this angle is D, so we I'll call this little D here. And this angle is R. So, I'll call this little r. Ah, doctor. Look at that. Dr. Sully, huh? Who would have thought? Uh, so the

11:37 sign of 62 would be what? In this case, it would be D over 12. The hypotenuse opposite because it's opposite 62. And you could use the 28. Either way is cool. Um, so we're going to solve that equation. And my other equation will be I'll use the cosine of 62. That'll give me what? That'll give me r over 12. So these are cool because these are great ones to solve. Uh they're they're I think the easiest ones to solve. If I have this and I want to solve for d, what do I do? I don't like fractions. Times that by 12 times that by 12. So these cancel. And now I know what side d is. I just type it right in the calculator. It's going to be 12 sine of what? 62. Boom. Done. And done. 10.59. Again, I'm not going to round. I'm just going to truncate. Cut that bad

12:23 boy off. Uh 10.59. And that seems reasonable for this triangle. And then over here, let's do the other one. Same thing. I'm going to multiply both sides by 12. And if I do that, I'm looking at those cancelling and r is going to give me what? We'll do the same thing, but now I'm doing the cosine of 62. And again, that's just some decimal. And it's going to give me 5.63. Perfect. And that's this side down here. So, I found everything. I found the angles. I found all the sides. I'm good to go. Just a word of the wise, sometimes you're going to get one. Let's get Dr. Sully out of here. That's kind of creeping me out. Uh, you're gone. Oh, he's still there. We can't get rid of can't get rid of the face. Sometimes you're going to be stuck. And I You're going to have to add this into your

13:08 notes here. Uh, just to see if you're you're with me here. So, sometimes I'm going to give you something like this. Here's a triangle that's like five and let's say eight. And you're going to have to find the angle. So you may have to set up something like, oh man, I need to find this angle. So I'm going to say uh what this is opposite over hypotenuse. I'm going to say the s of x is what? It is 58. And then you're going have to do your inverse thing to find an angle. And a shortcut if you want, you don't have to do trig for everything because it's a right angle. What else works? Pythagorean theorem. That always works for right thing right things right triangles. So a square plus b square= c^2. So you could in this case if you want to find the missing side you could say hey what is 5^2 plus some unknown

13:56 let's call it y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y^2 = 8^2 and then you could solve for that and find this side. So that's all the practice. I did it crazy fast. If you were just so lost on that, man, ask your teachers for extra practice. Go back, watch the geometry one. I'm assuming you've seen it. That's like kind of phase one of this section. Let's go to phase two. Luckily, phase two is fun. It's so fun. It's like an amusement park. It's a ferris wheel and it's short. So, I'm going to burn through this fast. So, we're going to keep looking at this ferris wheel because trig it's a perfect representation of trig function. So, the first thing I want you to do is check this out. All right. So, check out this ferris wheel here. We're going to look at this ferris wheel and you can see this red one is like, hey, when you get on the ferris wheel and what's going to happen is we're going to spend this chapter checking out your height over time. what happens as you ride this

14:41 ferris wheel as your height goes up and then back down and up and back down and so on so forth. But for now, look at this angle being formed. That's all I want us to get from this. I'm going to hit play here and watch it spin. Do you see the angle here in the middle? Boom. 90° angle. And then it keeps going straight across straight line. Boom. There was a straight line and then we're at that 270 and it's going to start back over here. So, we did a whole circle. Isn't that cool? So, I'm back. I haven't even passed it. So, this angle here is 48°. So, the idea is I want us to get used to is like, yeah. Oh, nailed it. Look at that. There's our 90° angle right here. I can see my 90° angle from the blue from where I started. But, we're going to need more than 90. So, right triangles are great. And they're going to come in again. But, we're also going to go above all the way to 180. It's hard. It looks like 100 was 180.

15:28 It's straight up and down. That's a straight line. Is 180. 190. 200. 210. What happens when I go all the way if I can hit it? 270. There we go. So, I went 90 180 270. So, these are angles we're going to need. And then we're going to bring it all the way down, finish it up to 360, and then it starts over again at zero. So, we're going to look at angles that go way bigger than what maybe you're used to dealing with, bigger than triangles, uh, for this. Pretty cool. Okay, so that was super cool. So, now we see those angles. So, let's take a look at some of these angles. And my kind my angle kind of fell off the screen here, but when you have an angle, uh, so our angles are going to go above 90 degrees and they're going to go above obtuse. They're going to go above 18. They're going to be huge. But make sure we remember things like vertex and an angle is formed by two rays. So here is our

16:14 angle. Um we're going to put them in standard position because we're going to go back around the circle around this ferris wheel. We're going to say if this is directly to the right is called our initial side and this is 0 degrees. So anything that would come off of uh in standard position comes off as 0 degrees and this is the initial side. So it's starting side. So, let's say over here I point up. Whoa. What kind of angle is that? Let's try that again. That was awful. Here we go. Terminal side. So, this is where my angle ends. So, I'm actually talking about this angle right here. This is the terminal side of the angle. Uh, it started here. So, basically for I know this is a right angle. So, from here to here is 90°. If I went straight across straight line, that's 180. If I added another 90 to that, that's 270. And then I'm back here

17:01 to 360. So, it starts over. So this angle is above 90 and maybe it's uh 30° past 90 or 60 degrees past 90. We're going to get some things like hey this is 150° angle. So we're going to start measuring angles like this like 100 roughly 150 degrees. Other things we need to know are quadrants. Just a quick refresher. This is the first quadrant. This is the second quadrant. This is the third quadrant. And this is the fourth quadrant. So I'm going to ask you what quadrant do angles fall in and things like that. So make sure you can name this what quadrant. So like this angle falls into the second quadrant. Pretty cool. So this is like laying the foundation of these trig functions um before we get to graphing them next thing, which is pretty great. All right, so let's take a look at this bad boy

17:46 here. Let's draw some angles. So the first one, 315. So if you need to draw this in here, I start at zero. I know that straight up is 90 because it's a right angle. Straight across is a straight line, 180. Add 90 to 180, you get 270. So that's a good reference point. So 315, what quadrant's that going to be in? 1 2 3. It's going to be in the fourth. You're past 270. You're not all the way back to 360. That would be a full circle. So I'm going around to something like this. 315°. Boom. Isn't that pretty cool? Can I go negative? Sure. If this is zero, I'm going to go negative 150. So this is like saying 90. I'm going in the negative direction. So if you go counterclockwise, that's negative. If you go around like a uh I'm sorry. If you go clockwise, it's negative. You go

18:33 counterclockwise, it's positive. So, it's weird. So, I'm not going to go this way. That would be positive. I'm going to go negative 90. Not all the way to the negative 180. Somewhere over here. And I'm going to say something like this. This is 150 degrees. Awesome. Very good. Can we do 460? Sure. Same thing here. I know I start at zero, but if I do a whole lap, whatever have I gone? I've gone 360. So, this is above 360. How far above? Well, it's 100° more than 360. So, I got to do one 360 plus 100 more. So, I'm going to get an angle that looks like this. Just past that 90 on the second loop. So, I could do multiple spins around the ferris wheel uh and get multiple answers there.

19:19 Awesome. And now, let's come down here and name some angles. So um I I want to name these from the standard position. So if I give you this is 120 and actually you could actually even call this angle negative 120 but this part's 120. Can I name this in the positive direction? Can I name it this way? So from the positive direction if this is 120 the whole circle is 360. So if I subtract them sure I can come up with this. This is 240° angle. So this angle is a 240° angle. Awesome. So just kind of getting comfortable with these. How about this? What if this chunk is 45? It's 45 less than what? Well, straight line's 180. It's 45 less than 180. So if I subtract 45 from that, what do you get? You get 135°. So that angle is 135.

20:09 So each one of these is like a little puzzle. How do I figure it out? I want to go from here to here. So how far did I travel? I went 90. I went 180. I went 270 plus a little bit 30 more. So, what's 30 plus 270? 300. Awesome. Super cool, man. Rock out. We got one more right here. So, can we do co-terminal? What are co-terminal angles? These are angles that all share a terminal side. So, what does that mean? It's different ways to label the same angle. So, we kind of had an example like this earlier. What if I called this this angle right here? I could call this angle uh negative let's say 140. Let's say it's negative. I just kind of estimate. So I went 90 plus a little bit

20:56 more. So let's say it's negative 140. How do I call this another way? Same thing in the standard form. What is it over here this way? Well, the positive way to call it would just be what? Subtract them. If you take your 360 minus your 140, what do you get? You get 220. So this is a 220° angle. Excellent. How about this angle? I'm going to make I'm going to make one up here. So again, if I got something like this, let's say this is 140 in the positive direction. So that's 14. No, that'll be too easy. Don't do that. Let's change it. What? Z. Hope I'm not putting you to sleep there. I got the Z's coming in. Uh, let's do this. Let's say this is a 130 degree angle. So I went past 90. I got to 130. So there's a positive way to

21:43 call this angle. What's a negative way to call this angle? Well, negative way would be to go this way. So two different ways to name an angle. I got 130. If I take the it's got to add up to 360. So if I take the 360 and subtract the 130, what do I got? I have 230. So I could say this is a -230° angle. They're the same. 130 and negative 230. The exact same. In fact, if you wanted to get crazy with it, I know this is 130. What if I would have done this? Let me get a different color here. What if I would have said, we'll get purple out for this. Okay, I just went right by it, did a whole loop, then came back to it. Like a whole spiral there. So, you could actually do 360 plus 130. Add those together, you get 490. That's the same

22:30 thing, too. 490 is the same angle as 130, which is the same angle as -230. It's all the same angle. It's just different ways of nailing it. You could do another 360 to that if you really want to add another 360. It did in 50 uh and then be what 850. You could add another 360 and do another lap to name it. So there's infinite amount of ways to name it. There are infinite co-terminal angles. They're all co-terminal. They're just different ways to name it. Awesome. So that brings up a really interesting point. What if I give you a 30° angle here? This is a 30°ree angle. And I say, "Hey, you need to name every co-terminal angle possible." You can do it, but we can't sit here and list them all because yeah, 30 is cotminal to what if I did 360 plus 30? What if I did 360 plus 360 plus 30? What

23:16 if I did a whole another loop? So, there's a ton of them. So, how do I write it? Well, I can do something like this. I can say I have a 30°ree angle plus I'm going to add any kind of combination of 360n. What is n? This is where n my handwriting's bad. n is an integer. Remember what's an integer? It's just a positive or negative whole number. So this is the number of loops you do. Maybe you do zero loops. If n is zero, that's just 30. If n is one, you do one loop. That's 390. If n is two, you do two loops. 720 plus a 30, it's 750. So it works infinite amount of loops. You can even do negative loops. Put -2 in there. This will give you every possible co-terminal angle in the world. So if we got to find them all, we're going have to write a little little expression

24:03 here. If we're just doing positive and negatives, we can list one. That is it. I think the best way to do these is just to do them. H is that good advice or what? Practice these. They'll get easier. I know maybe weird at first. Uh check out this pretty cool manpowered ferris wheel here to wrap it up and good luck on the match check. Peace out.