SENO, COSSENO E TANGENTE - TRIGONOMETRIA NO TRIÂNGULO RETÂNGULO \Prof. Gis/

Hi guys, welcome to another class on Gis' channel. And today you will learn trigonometry in the right triangle. Let's go?

Take the opportunity to subscribe to the channel and leave your thumbs up for Gis! And let's talk about trigonometry. First, let's try to understand the meaning of the word trigonometry.

So trigonometry is the combination of two Greek words trigonon which means triangle together with metria which means measure, so translating right we then have the measure of a triangle. And as I brought you here, trigonometry is the part of mathematics dedicated to the study of metric relationships, okay? Existing between the sides and angles of the triangles of a rectangle, okay?

So for you to better understand the initial concepts, I brought you a rectangle here, right? What if we divide this rectangle in half like I did here and we get two triangles that are called right triangles, because they came from a rectangle, okay? And what do you know about the characteristics of a right-angled triangle?

You can talk? A right-angled triangle then has an angle of 90 degrees, right? And the other two angles that we have here, this angle with this angle, together, the two have to add up to 90.

So this means that these two angles here are two complementary angles because they add up to 90° degrees because you need to remember the sum of the internal angles of a triangle is 180. If it is 90, then the other two together also have to add up to 90, ok? So what have you seen about metric relationships, I said metric relationships in a right triangle?

Very good for you who answered the Pythagorean theorem, so if we analyze this right triangle here, what do we have here in the right triangle? Here we have the hypotenuse, what is the hypotenuse? The hypotenuse is the side opposite the 90 degree angle, it's the longest side of the triangle, okay?

And the other two that we have here are called cathetos, so there's the cathetus here and I have another cathetus here. Well, so far so good because you've already seen the Pythagorean theorem, and what is the Pythagorean theorem really like? Just to remember.

Is that really what? Leg squared plus leg squared, and this results in the hypotenuse squared, so just to make it short for you, I'm going to put it here as A, here as B and here as C, okay? For us to write the Pythagorean theorem.

So the Pythagorean theorem is B squared plus C squared which will result in the hypotenuse squared, so for those of you who forgot, a reminder for you of the Pythagorean theorem, ok guys? And so now let's continue, right, what's the difference now that we have Pythagoras' theorem, what I just said here, to when I'm going to study trigonometry in the right triangle? Did you carefully observe what is in the definition?

It's the metric relationships between the sides, because here I only did two sides, but now I'm also going to relate the angles, okay? So I'm going to use this other triangle here as a base, so again, just to mark the right angle and the other two angles, right, which together are 90 degrees. There again, so you've seen it well, here I have the hypotenuse and here I have the two sides, I'm just going to abbreviate it now, okay?

Which we have already written here in full. But now folks, we then have to relate the sides to the angles, and relating the sides to the angles, and I will have the so-called opposite side and I will have the so-called adjacent side, my God, what is all this ? So you see, it will depend on who will be the opposite side and who will be the adjacent side, it will depend on the position of the angle, the angle of view, where I am based.

So let's pretend that I Giz is based on this angle and I'm going to mark G de Gis here, pretend that I'm based on this angle here, if I'm based on this angle here, this leg will be what for me? He's going to be my opposite side because he's on the opposite side of my viewing angle, okay? Opposite side, so we call this the opposite side, okay?

Opposite Cathetus. Ok. Now this side that is next to my angle of view is stuck there in my angle of view, it will be my adjacent side.

Look carefully then to mark, opposite side okay, and adjacent side. But what about the hypotenuse, the hypotenuse will always be hypotenuse, it won't change depending on the viewing angle, why? Because the hypotenuse will always be facing our right angle, okay?

But Gis, you must be asking now, what if I changed the angle of view like the one here now, would the position of the peccaries change? Then it would change people, if you looked at this angle here this one now wouldn't be the adjacent side but it would become the opposite side and this one here would be the adjacent side then, if I looked at this angle, but we are taking it as a basis Is this angle here? So that's why this differentiation, you will look at the triangle and see which angle is highlighted, right, because the exercise will always highlight an angle.

And then you will use it as a basis to find out who is the opposite side and who is the adjacent side of that respective angle, right? Now I'm going to explain in more detail to you how we do the applications, right, with the relationships with the trigonometric ratios. Let's go then?

So you should check that these trigonometric ratios that I'm going to mention now, they will depend on the angle that I'm considering, okay? And like that time in the previous example I mentioned this angle of view, right? Now I'm going to start from this angle of view.

So considering your, our angle of view, I'll have, remember, there's a side and the longest side of the triangle, the hypotenuse, that's it, you remember Pythagoras' class. But now , as we have to relate the side to the angle, this side here will be the side, which is it? It's facing my angle of view, which will be called the opposite side and the other one that's next to it, what's left is the adjacent one.

So write it there, adjacent, ok? Once this is done , you have already identified it. Then we will be able to establish the trigonometric ratios, which are three trigonometric ratios: sine, cosine and tangent.

Three names: sine, cosine and tangent, which are the three trigonometric ratios. But for what reasons? Reasons so it is indicated through division, so guys, when I want to obtain the measurement of the sine, the sine of an angle C I will mark the sine of the angle C, okay?

So who will be the sine of angle C? Every time I want to find the sine, I will then measure the opposite side, so I will abbreviate the opposite side by CO, so it is the measurement of the opposite side divided by the measurement of the hypotenuse. I'm abbreviating it, folks, so I don't have to write it all the time.

So the measurement of sine C, of ​​angle C, will be the opposite side of the hypotenuse, okay? So it's always like this, sine, opposite side and hypotenuse, mark it there for you to memorize. When I'm going to talk about cosine, there's one more thing I didn't say, okay, the abbreviation for sine is sin, it's like this, the abbreviation for cosine is cos, cos, so here it will be the cos of angle C, which means cosine from angle C, okay?

And the cosine of angle C will be the ratio between the adjacent side, which is this one over the hypotenuse, ok? And finally, if I do the tangent, the abbreviation for tangent is TG, okay? The tangent of angle C will then be equal to the ratio of the opposite side to the adjacent side.

Guys, remember that these are our three trigonometric ratios, now when we have a triangle and we have the angle in it, we will use these three ratios, the three measurements, and then you need to memorize them. A way of memorizing that I learned, I always pass it on to the students, oh, I think here, oh, pretend it's a river, I ran, I fell and had a coke, so the tangent will always be the coke, okay? Make a relationship, right, so we don't forget.

I ran, fell and had a coke. Sine, cosine and tangent, right? But you might also be wondering, what if I changed the viewing angle here?

Then it changes, here it would be the opposite side and here the adjacent one. But this here, people, is always the same thing, there the sine opposite the hypotenuse. Then you would look at the opposite side of here and it would move over here, okay?

So now I'm going to do the application, I told you about these measures and for you to mark them and now we're going to do the application, solve the exercise by applying this, okay? Let's go then? Determine the sine value of Beta, Beta here is this letter which is a Greek letter, okay?

Beta cosine of Beta and tangent of Beta, okay people? So look at the right triangle, just remembering that this trigonometry we are applying is valid for the right triangle, okay? And here the Beta angle is here the viewing angle.

So let's go, calculating the sine of this angle Beta, do you already remember what the measurement of the sine is , sine is the ratio between the opposite side so I say that it is always important to go back and write so as not to forget, opposite side by the hypotenuse. So what does this mean that the measure of this angle Beta will be? Now comes the secret, where is each thing here in this triangle, right people?

So if you want, firstly, you will identify in the triangle what each thing is, so look, we already know that that longest side is the hypotenuse so I'm going to write H here for hypotenuse. Then I have to identify who is the opposite side and who is the adjacent side, and the adjacent side is the left side. Who is the opposite side?

Is it 15 or is it 20? Look where my angle of vision is, the angle is here, this side here that is facing is opposite, so here is the opposite side, and here, what is left adjacent side, I'm abbreviating it now because it's always abbreviated, okay? good?

So opposite side adjacent side hypotenuse, now just apply it here, so that means my sine here will be the opposite side which is 20, 20 over the hypotenuse which is 25 and then I can come here and make a simplification, here we We can simplify by 5 and we will find that here it will be four-fifths, so that means that the measure of the sine of that angle is four-fifths, OK? Now let's do the same with the cosine, cosine of the angle Beta = side adjacent by the hypotenuse, don't forget these reasons, then it will be side adjacent, I already identified that it is 15 so it will be 15 by the hypotenuse which is 25 and once again I will divide by 5, dividing by 5 here by 13 and here by 5. So it will give three fifths, and now to obtain the tangent of this angle I do the opposite side, which is Coke, right, by the adjacent side and opposite side is 20 and the adjacent side is 15 then simplifying by 5 again I will find 4, 15 by 5 are 3, four thirds.

And one more thing that I'm going to tell you, did you know that if I take the measure of the sine of this angle, which is four-fifths, look here, four-fifths, and I divide it by the measure of the cosine, which is three-fifths, look here what happens, dividing two fractions, do you remember the ping-pong method? When I divide two fractions, the quickest method is, I multiply this one with this one which will result in 20 and I multiply the 5 with 3 which will result in 15, plus 20 15ths still doesn't give anything. Let's simplify 20 15ths, you can simplify by 5, right?

By 5, and this will be four thirds. Can you now relate this measurement I found to anything here? Here is the tangent, guys, so it means that when I divide the sine of that angle by the cosine of the same angle, okay, I can't divide it by another one, it has to be the same angle, and every time I divide the sine by the cosine I get the tangent Oh, look, then divide this one by this one and I would find this one, so it's already a trigonometric relationship that we'll talk about later, but I'm already telling you about it in this class here, OK?

Now I'm going to do an exercise that's a little more difficult, so this one was very easy, right? Let's go, now see that I brought an exercise within a context and not a simple exercise like the one we did, that was a "little workout", right, the warm-up to do these others here, okay? So see that trigonometry was invented out of necessity, it was invented, it was created, you know, out of the need to measure distances that are inaccessible, because imagine in this situation where the plane here is taking off, I'm not going to take the ruler, the measuring tape to carry out this measurement, right?

It won't make sense, just like when I want to measure the distance from here to a lake, for example, right? We use devices today, right? They have all the technology, right?

In the past, these instruments were very rudimentary that they used, but there was always this relationship between the sides and the angles of a triangle, okay, the angle is what I always look at one specifically, well then, let's go? What matters is that you want to exercise, right? A plane takes off at an angle of 18°, okay?

So how high is the aircraft after traveling 35 km maintaining this same angle? So my suggestion, as an initial strategy, is for you to create an outline of what you're dealing with in this statement and thinking about the outline, you know, when I deal with trigonometry in a right triangle, it will always form a right triangle. So I already got my triangle ready here for you, so there's a triangle here, it would be our horizon line, right, which is parallel to the ground, my little plane left that point and my little plane lifted up, look at my plane, how beautiful, it took flight right?

Under an angle of 18 degrees like we have here, it takes off under an angle of 18 degrees so the ground forms the horizon line, the little plane takes off and then what does it do? Then he traveled 35 km, so it would be 35 km here on the hypotenuse. And the statement asks, how high will he be when he has covered 35 km?

So I can put x and I want to determine how high it is after 35 km traveled, right? Keeping that same angle, right, the statement specified this well. And now how am I going to solve this?

I've already made the sketch, I've already entered the values ​​I have, now we're going to identify what measurements I have there, opposite side, adjacent side hypotenuse and then? You can already know one right away, right? Remembering that here, as is the height, the height formed an angle of 90 degrees here, so this 35 will be my hypotenuse, so I will mark it as H, okay?

And what more information do I have on this side x here he is a guy, but have you already identified which guy he is? Is it adjacent or opposite? Look at the viewing angle, I'm taking the angle of 18 degrees as a basis.

So here I have the opposite side, I'll mark it for you. And this one would be my adjacent side. More people, this is where many students start to get confused in solving the activity, because they want to say that this is the adjacent side, but I'm going to tell you something, there is some value here, you want to discover something about this , adjacent cat doesn't want it so it doesn't need to, let's make things simple, okay?

So I have opposite side with hypotenuse. What trigonometric ratio forms the relationship between the opposite side and the hypotenuse? You already know, right?

Very good, for you who answered what forms the sine, right? Opposed side by hypotenuse and then the sine of this angle here, is the sine of 18 degrees which will be equal to the opposite side which is x by the hypotenuse which will be 35. But how do I finish this exercise now?

So guys, look here in the statement, he is giving me what the sine of 18° is, he gave me this, and commonly if you are doing a test, an activity that doesn't, you don't have access to a scientific calculator that also determines these values ​​for you, the statement will be similar to the one I brought you, okay? So the only secret will be for you to find out if it will be sine, cosine and tangent because I see that the advertiser gave you the three measurements here and if you don't know how to identify which one you are going to make a mistake, okay? So I have already identified that it is the sine and the sine of 18°, it is worth 0309.

Now we come here and put 0309 = X by 35 so I see that, now here I have a proportion, right, so what do we do? You can put 1 down here and cross-multiply, then I'll do 1 times x, I'll finish the math here, okay? 1 times x which is x = 35 x 0309 and doing this multiplication we will obtain 10 integers and 815 thousandths, and what will this answer be given in?

In km. So you want the height of the plane when it has traveled 35 km at an angle of 18° it will be 10. 815 km high, right guys?

So you saw this exercise, it's an application exercise, okay? It involves a context so it wasn't that difficult, right? Because you already know how to identify sine, cosine and tangent.

Now we're going to do another one. This example now looks like this, at the top of a building, so I have my building here, I brought the sketch, right, because always the strategy for resolving these types of issues is to make the sketch first OK? A person observes a car parked 80 meters from the building, so that means the distance from here to the car is 80 meters, okay?

And the person is right here, right, right in the building where the little person is, right? So I'm not going to draw a person there because otherwise it will look like I have to measure the person too. So she's observing this car that's there on the street, right, which says it's 80 meters from the building, she's seeing this car from a 40 degree angle, so finishing my drawing that I didn't finish, I'm a good drawer, guys, I I'm going to call the top of the building here in my car where my little person is, okay?

So does this mean that the angle formed is 40 degrees? So that's why it's important to read the statement well and get the correct information because if I put 40 degrees here I've already done the exercise wrong, right ? So the question asks how tall is this building?

She wants to know how tall she is here, okay? So can you visualize the right triangle I have there? He's here, right?

Right triangle is right here. So what measurements do I have in this right triangle? I'm 80, what's that again guys?

This 80 is a side, right, it's a side but it's the opposite side because it's facing my angle considered my angle of vision. I don't have anything here on the hypotenuse and I don't even want to know that measurement so I won't write it, but I have this measurement of the X which is the height of the building, at this height I need to find out it is considered my adjacent side. So I have opposite and adjacent sides.

What is the trigonometric ratio that forms the relationship between opposite and adjacent sides? Oh coke, coke is the tangent, right? So here I will have the tangent of 40 degrees which will be equal to the opposite side which is 80 divided by the adjacent side which is x ok?

And then I'm going to change the value, as I said in that other example, the statement gives me the values, so I just need to know which one it will be, then I'm going to use the tangent, so instead of the tangent of 40 I'm going to write 0839 = 80 over x then I fell again, right? In that proportion, put 1 below and cross-multiply. And then it will be x 0839, so let's put it here, it will be 0839 x = 1 times 80 which is equal to 80, so here I will obtain that x = 80.

As this 0839 is multiplying, it will be divided. And then dividing 80 by 0839 I will get 95. 35 meters which would be the height I was looking for here for this building, right?

So one more application where we did the trigonometric ratios and now we did the tangent. And let's go to the last example? Guys, see that in this exercise it says like this: the diagonal, so the diagonal is this one right here, of a rectangle it forms an angle of 25° with one of its sides, so see that because it is a rectangle it has an angle of 90 degrees here, and I have already identified the angle formed by the diagonal, okay?

Knowing that this diagonal measures 3 cm, so I'm going to separate it here into two triangles, just so you can see it better, here I have the measurement of 3 centimeters, which is the diagonal, determine the measurements of the sides of this rectangle. So here I'm going to call x and this other side, it's going to be Y for me, but guys, to solve this case I'm going to have to do two calculations, so I'm going to identify you, it's going to be sine, cosine, tangent. So let's start by finding out the value of X, so for now I'm going to delete that Y there so as not to get confused, okay?

So , based on this angle here of 25 degrees, the 90° and now when I relate the adjacent side to the hypotenuse I form which measurement then? I have the cosine measurement, cosine of 25 degrees. So see here in the statement it gives me the sine, the cosine and the tangent, that's why I need to know how to identify who I'm talking about, right, I need to find out that I, starting with the cosine of 25 degrees, will have x, which is the adjacent side by the hypotenuse here is 3, then I already know the measure of the cosine which is 0906 which is here = x over 3, then I can put 1 here and cross multiply it, okay, so it will be 1 times x which will be equal to x is equal to 3 times 906 so but it will be 2.

718, right, so this here will have the measurement in centimeters on this side of our rectangle, okay good? Now have I finished the exercise? No guys, because now I need to know the measurement of that Y that I had deleted and which measurement does that y correspond to?

It's the opposite side, so it's CO, and then we can see that now I can do CO with H which will be the sine or I can do Co with CA which will be the tangent so you can choose, but I need to use this here then it will be the sine or tangent, and then you identify the one you want to do Ok? I'm going to do the sine here then, then I have to do the sine of 25 degrees which is equal to the opposite side which is Y by the hypotenuse which is 3, then I already know that the sine is this measurement here of 0. 423 = y over 3 And if you did it on a tangent, you'll get the same result, okay?

And then I can put 1 down here and I cross-multiply it, so it will be 1 goes up 1, 3 times 0 = 0 then I count three decimal places so the measurement of y will be 1. 269. Look, I forgot centimeters, right, I have to enter the unit of measurement that I entered, here I entered 2.

718 cm, okay? And then guys, if you also wanted to solve this type of exercise, right, in addition to the tangent you could have done, you could, at the end here, to discover the measure of y, have applied the Pythagorean theorem, because see, you have the measure from this side, which is our 2718, you have the measure of the hypotenuse, and you need the measure of the other side, if you applied the Pythagorean theorem that would also work, it would be a little more complicated, but it would also work so you could do it by tangent, by sine or by theorem of Pythagoras to discover this last measurement here, right? So you saw how many possibilities there are, just interpret a statement well, people, which he always has in these sketches, so I replied he wanted the measurements of the sides of the rectangle, so we managed to find them.

And that was our class today, I hope you liked it, that you understood, right guys? There's no point in liking it and not understanding it. As long as you understand the explanation, I made it as detailed as possible so that you can be successful in your activities.

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