G'day, welcome to the TechMath channel. What we're going to be having a look at in this video is trigonometry. So sit back and learn all about it.
And if you like the video, please remember, hit the like button beneath the video there and subscribe to the TechMath channel. So trigonometry deals with this particular shape here, a right angle triangle. And what it does is it's a branch of mathematics that studies the relationships between the sides of this triangle and the angles that occur within it. Okay, so pretty much we can use say an angle here and a side length to work out other side lengths or we could use say two side lengths here to work out unknown angles. That's what trigonometry allows us to do.
So how does it do this? Well it's fairly simple. If we were to consider say an angle here in this triangle. So I'm just going to put this down and this angle here is called theta. Pretty much what it's saying is this.
For this particular angle here in a right angle triangle in this particular location in that right angle triangle. These two side lengths here would have a particular ratio. They would always be an equivalent length compared to one another. Okay, this length and this length would have a certain ratio, and this length and this length would have a certain ratio.
And trigonometry uses this to be able to work out unknown side lengths and unknown angles. So how do we do this? Well, the first thing we have to do is we have to be able to label the sides of this particular triangle. So in this particular triangle, you're going to notice we've got a right angle here.
We have this angle theta, which we've already labeled here. We also have three sides here. We have this longest side here. The longest side is called the hypotenuse.
I'm going to write that in, the hypotenuse. I'm going to put that down as a H. Here, we have the opposite side.
I'll write that over here, the opposite. What do I mean by that? This particular side is opposite theta. We put that down as an O. Along this particular side, this remaining side, which is next to feeder, we have the adjacent.
Adjacent, okay? Adjacent means next to, and we label that with an A. So now we've done that.
As I was saying, all these side lengths here, the opposite, the adjacent, the hypotenuse, all have particular ratios to one another based on whatever this particular angle here is. Okay, so there's three different functions we are thinking about when we're thinking about these ratios. because we have three different ways we can compare the sides. We could be comparing these two sides to theta, or these two sides, or we could be comparing these two sides. And our three main trigonomic functions are as follows.
We have the sine function, which is the ratio. of the opposite and the hypotenuse. We have the cosine function, which is the ratio between the adjacent and the hypotenuse. And we have the tangent function, which is the ratio between the opposite and the adjacent function.
Now, there's a really, really easy way we can remember these when we're doing these. And this is as follows. I'll write this mnemonic down right now.
And here it is. Some old hags can't always... hack their old age okay SOH sine equals opposite over hypotenuse can't always hack cos equals adjacent over hypotenuse their old age tan equals opposite over adjacent so when I was solving a trigonomic equation pretty much the very first thing I'd do is what we did first off here I'd label these unknown sides the next thing I'd do is I'd determine which trigonomic function I was going to use so we're pretty much all set now to solve some trigonomic problems.
So let's do that. So for our first example here, we have a right angle triangle. Okay, it has an angle of 35 degrees, it has one side length of 12 meters, and another unknown side length, which we're going to be trying to work out. So the very first step to work out this unknown side length is we are going to do what we do with any trigonomic equation or any trigonomic problem, we are going to label the unknown sides.
So first off, we have this long side here, which is the hypotenuse. then we have this side which is opposite this angle opposite this 35 here this is the opposite so which of our trigonomic functions deals with the opposite and the hypotenuse and you're going to see that it's sine here sine is equal to opposite over hypotenuse some old hags so i'm going to write this down sine theta is equal to the opposite over the hypotenuse and now what we do is we just go through and substitute in our values as sine theta this is sine 35 degrees is equal to the opposite. The opposite is what we're trying to work out here, x. So I'll put that in as x over the hypotenuse, which is 12. So we can now work this out a little bit further. We could actually say, okay, sine 35, we put that into a calculator.
We're going to get the answer as 0.5. 7, which is equal to x over 12. What can we do now? So what we have to do is we have to get x by itself.
Okay, so x is going to be equal to what? There's a little trick I use here. This may or may not help you. You may or may not like it. Okay, I'm sure I'm going to get plenty of hate for this.
But what I do is this. When I'm not certain what to do here, and I'm trying to solve this particular problem here, I just write up an equation next to it, a friendly equation as it were. The equation I'm going to write is this one, 3 equals 6 over 2. And we're trying to deal with this particular value here, the value up here. So what would you do with 3 and 2 to get 6? Well, you'd multiply them.
So we're going to multiply these two numbers, 12 times 0.57. So 12 times 0.57, and we'll get our answer. So if you do that, what answer do you get? You get our answer of 6.8.
8 meters. Okay so this side length this opposite is 6.88 meters and that's how easy trigonometry is to use. Okay so we're going to go through another example and then I'm going to go through an example where I look at how to work out the angle from 2-9 side length.
So it's a bit of a tweak here so stay tuned for that one as well okay. But let's just go through another one of these time examples. Okay for our second example let's have a look we have a right angle triangle we have an angle of 48 degrees. We know that this side length here is 15 and we're trying to work out this unknown side length here.
So let's label our sides first. We have this particular side here which is opposite the angle here so that's the opposite. We know that this one here is the hypotenuse that's the easy one to spot so it leaves this one here being the adjacent okay and it makes sense it's the shorter one that's running next to the angle here.
So which one of these functions uses opposite and adjacent, you're going to see here is tan. Tan theta equals opposite over adjacent. So let's sub in our values now.
So tan theta becomes tan 48 degrees, which is equal to the opposite, which is 15 meters over our unknown, our x. Okay, we can put tan 48 into the calculator. If you do this, you're going to get this answer of one.
0.11 okay the opposite and the adjacent have that particular ratio of 1.11 for an angle of 48 degrees which is equal to 15 over x so now to solve for x and if you're not certain what to do you might know this straight away but you could do this once again you could go okay 3 equals 6 divided by 2 and we're trying to work out the value on the bottom here the 2 so that would be 6 divided by 3 this number divided by this number this number divided by this number x x here is going to be equal to this number divided by this number, 15, divided by 1.11, which is equal to how much? 13.51 meters. Okay, so that's how that particular type of R function in trigonometry works. It's pretty simple, right? Now we're going to go through some examples.
We're going to look at how to work out the angle from known side lengths. It's fairly simple. There's just a couple of tweaks with this.
So in this example here we have a right angle triangle and we know two side lengths. We know that this side length here is 105 metres and we know that this side length here is 33 metres. What we're trying to find out is we're trying to find out this unknown angle that would accommodate these side lengths.
So how do we do that? Well it's just one little variant and I'll get to that as we do this particular problem. The very start though is exactly the same. We are just going to go through and label whether our sides are opposite, hypotenuse or adjacent.
So we know this long side here is going to be the hypotenuse. We know that this side opposite angle theta here is the opposite. So which one of the functions are we dealing with? This is our second thing we could deal with, which function? And we're going to be dealing with sine here.
Sine theta is going to equal the opposite over the hypotenuse. So what is the opposite over the hypotenuse? We're going to see here.
that we have the opposite which is 33 meters over 105 meters okay so sine theta is equal to 33 over 105. if we were to work this out what's 33 divided by 105 you're going to see that sine theta is equal to 0.314 okay that's just a matter of going 33 divided by 105 and we get this answer here so what we do now is just a little variant because we have to actually go back. We've got the ratio. We're trying to go back to the angle and you're going to notice on calculators that there's either a second function or something like that, that allows you. that go from sine to this particular thing, sine negative one, okay?
We want to be using that here. We're going to be hitting 0.314, and we're going to hit sine negative one, or second function sine here. If we do that, we're going to get the answer of theta being equal to 18.3 degrees, okay? So just make sure you know how to do that on your calculator, okay?
Anyway, I'll go through one more of these examples. Okay, for this example here, we have a right angle triangle. We have two side lengths. We know that this one here is 17. We know this one here is 12. And we're trying to work out the angle that accommodates these.
So let's go through and do this. The very first thing we do is we're going to label our sides. We have the hypotenuse, which you can see.
We're not going to be dealing with the opposite. We're in fact dealing with the adjacent. So which one of these functions are we dealing with?
And you're going to see here, The adjacent, the hypotenuse is the cosine function. So cos theta is equal to the adjacent over the hypotenuse. Okay, so what is that going to be? Cos theta, which is what we're trying to find out, is equal to the adjacent, which is 12, over the hypotenuse, which is 17. So we work out what 12 divided by 17 is. We get the answer of 0. 0.71.
Okay, cos theta is equal to 0.71. So we're going to be not working out cos, we're going to be working out the inverse of cos, cos to the negative one. So you're going to hit second function, cos, and you're going to get, when you do that, you're going to get the answer for theta. Theta, or our angle here, is equal to 45.1 degrees. So that's how you go doing trigonometry at its most basic.
It's pretty simple, right? It's just those tweaks there and it's also getting to know the calculator that you are using. So anyway, hopefully that video is of some help to you. If you've got any problems, please let me know and I'll make some more videos on trigonometry.
I'm sure I'm going to have some issues where people are going to get stuck with these. Please, if you like the video, remember, like and subscribe. Hey, and below the video in the description there, there is the Patreon feed there.
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Bye.