All right in this section here we're going to talk about trigonometric functions of a trigonometric triangle. And the first three I want to talk about is sine. Si and E, cosine, and then tangent. Okay, these are the abbreviations right here. So sine si n cosine CEOs, tangent ta n. Now notice we're talking about the sine of theta, that this this little symbol means data Our angle and we're talking about this angle here. So we're talking about the sine of theta, theta represents an angle.

And at this point, we don't know what it is. All right, cosine of that angle and tangent of that angle, these numbers and I'm going to go through them in a minute. But basically those numbers there are just a ratio of the sides. All right? So let me stop here and clear up the scuzz. Slide.

Okay, I've cleared off the slide. So let's, let's look at the first one here. The sine of theta. All right, the sine of theta is the opposite side. Over the high plot news. The opposite side of theta is theta.

What's my opposite side? It's a isn't it isn't side A Opposite of theta over the high pot noose. And we always know that C is the high pot news. So the sine of theta, which is a ratio in this example is going to be, we can say a oversee. Now I've, I've given you some values as four and C is five. So if I divide four by five, I get a ratio of point eight.

All right, so the sine of theta is 0.8. Okay, now let's go on and look at cosine, we want the cosine of theta. And if I look at that it's the adjacent side over the high partners will last theta, whereas the adjacent side, isn't that be right here. Isn't be a Jason To sine theta. Yes it is. And again, we've assigned three for B, we know that the high pot news is five.

Okay, so three over five is zero dot six. All right, and and these are just units. Just to stop here with these just to present this, this illustration here that we have four units, three units, five units, I didn't say what the units represent. We're just trying to get a handle on what these mean. So let's do the last one, which is tangent. Okay, so now we're looking at the tangent function and I did clean off the screen and I made a correction here I forgot to put those in.

So I made a correction I put them in so the tangent function is the opposite side over the adjacent there's a deck is my opposite side is my adjacent side. over B, A equals four, b equals three, I do the math and I get a ratio of 1.33. All right, now. Okay, so we did that we know there's a ratio. So why did we do that? Because if we do that, we can use another trigonometry function, which is the sine minus one, where we can actually find that angle theta.

And again, what we need to do is we can use the calculator and I'm going to get mine down. All right, let's move it over here. Let's Well, let's, let's see, let me just stop here and Okay, so make sure you're in scientific mode here. Again, this is the calculator which is part of the operating system. And so we found, here's my sine, cosine and tangent. But look at what I'm saying sign and I've got a minus sign.

All right, so I hit the up key, I'm going to get the reverse sine function, that's what that minus one means reverse. When I did the sine, we will looking for the ratio of the sides. Now what I'm looking at is, okay, I'm giving you the ratio of the two signs. What will the angle B. So for instance, my ratio here is point eight. So what angle here will give me a ratio of point eight with the opposite side, this one in the high pot news?

Well, let's see. So all I do is I've got a little function button here. So I can change my function. So I'm making sure I see sine minus one. I'm going to plug in zero dot eight. And I'm going to hit this.

And it's 53 dot one for the most part, which I show you right there, I rounded off to 53. All right. So basically, what we know is, when I have a sign, when I have a rep sign gives me a sign function gives me a ratio of point eight, because the length of the opposite side and the high pot news, then angle is going to be 53 degrees or approximately 53 degrees. All right. Now, I didn't show it to you. But we can also do the same with the, what we call the reverse functions, cosine and tangent.

So if I go Go through these I should see 53 degrees or approximately, because they are repeating decimals so they're not going to be. So the numbers to the right of the decimal point I've got to be exact, but I should see 53 degrees point something. All right, so let's clear the calculator. And I gave you a cosine of zero dot six. Okay, so zero dot six, we're going to use the reverse cosine function. And then again, I get 53 dot 130.

That's exact. Okay. Let's clear the calculator. And now we'll look at the reverse tangent function. In this case, the ratio of my tan on my tangent was one dot three, three. So one dot three, three, hit the reverse tan function.

And then again, I get 53 degrees. So we all check. All right, and that's a way to find the angle theta. Now, why is that important? Again, when we look at AC analysis, okay, we can represent reactive properties and electronics by my adjacent side and by my opposite side, okay, and again, I've got one to show you why maybe the next slide or two, I don't really want to get into the theory of electronics right here. This is not the place for it.

The only thing I want to do is explain to you one of the reasons why we use a right triangle. And that's probably the main reason is to look at complex what we call complex impedances in an electronic circuits that represent resistive and a reactive element. And again, that's it for that right now. All right, so let's play the slide and go on to the next one. Okay, here's a slide that I spoke about, I thought there was one one down, but that's fine. Right here we're looking at at a electronic circuit or impedance and we use the right triangle to find the resultant z.

On the side of the triangle, let us see are the high pot news. So what I give you here is I give these sides a name. So for instance, a, we're going to call X sub l. And that's measured in reasons resistance and resistance is measured in ohms. And all On the this side here, the adjacent side, because there's my angle theta. Okay? This is the resistance that is also measured in ohms.

And again, I'm not going to go into the difference between AR and XML here. That's not I'm just giving you wedding, just giving you a, a taste of AC circuit analysis. Okay, that's all I'm doing. So what when I use Pythagorean Theorem, I can find z, which is the result of these two properties. All right. So over here, we start giving some numbers we say okay, X sub l, which is on this side here is six ohms.

And our Which is on the IR axis, or the adjacent side is eight ohms. And I want to find z. Well, we know from Pythagorean Theorem. If we squared A squared plus B squared, and go through the math like we have, we're going to get the answer. So this is what I show you over here. So six squared plus eight squared and I just go through the steps six times six is 36. eight squared is 64.

If we add that we get 100. The square root of 100 is 10. So I know that z right now was 10 ohms. So if I have an XML of six ohms, and an R right here of eight ohms. When I go through the map, and use Pythagorean Theorem, I find the resultant of 10 ohms The other thing that we would like to know to again in in AC circuit analysis is what we call the phase shift. And the phase shift is represented by my angle theta right here.

Alright, so now we can use the sine function which I talked about in the last slide, we can say six over 10. Okay, and basically what we do is we get a point six. And then we use the reversed sine function with our calculator, and we find an angle of 39 degrees. So what we're saying here is I have a z or a complex impedance of 10 ohms and I have a phase shift between voltage and current in this AC system. circuit of 39 degrees. And again, we'll go into that.

What does that mean? What's 39 degrees, all that will do that when we get there. All right. So that's what I'm trying to show you here. I'm just trying to give you something to hang your hat on to understand what we've done. Alright, so let's go on to the next slide.

All right, on this slide, I just like would like you to do a couple of problems. I've given you a couple up here, on this one here, x sub l equals 10 ohms r equals 10 ohms. Find z. Okay. And then angle theta i, and then just another one here. As we change the values here, XML is six rs 12 and z equals what?

Find the angle theta here and see if we can Take the time and do it if you need to go back to the previous slide, look at that. Again to do these the informations on that slide the way I presented it, and on the next slide, I've given you the answers. All right. Okay, here are the answers. They're right here. Here's my angle for the first one, answers for the second one, and here's my angle for the last one.

Okay, if you didn't get it, you got every slide, I give you a contact number, send me an email through this website or through this platform. I'll help you out. And this pretty much ends. This, this lecture here. I see you in the next one.