34 Funciones trigonométricas II

We have seen in the previous video the behavior and representation of the functions sine and cosine of x, so the one we need to complete these elementary trigonometric functions would be the tangent of x, which is the one we are going to see now.

So once again we draw the coordinate axes, as you can see, graduating the horizontal axis from 90 to 90 degrees. We are going to make our table of values and if before we have used the goniometric circumference to see the values of sine and cosine to help us, now what we are going to do is help us with this formula that is relating the three trigonometric reasons. We already know that tangent is sine divided by cosine and since the values of sine and cosine are simple, so we are going to find out the values

of the tangent for different values of x. I explain myself, for example, when x is 0, the tangent of 0, if you don't know it by heart, is like the sine of 0 between the cosine of 0. The sine of 0 is 0 and the cosine of 0 is 1, so 0 between 1 is 0. Then the tangent of 0 is 0, and we have this first point of coordinates 0, 0.

When the angle is 90, the tangent of 90 is sine of 90 between cosine of 90. The sine of 90 was 1 and the cosine of 90 was 0. So here I have 1 divided by 0, which, be careful, does not give 0 and does not give 1. If you try it with the calculator, 1 between 0, what will happen is that it will give me error.

What does this mean in practice? You can't divide a number by 0 and what I'm going to find here is a result that tends to plus infinity or minus infinity. According to the denominator, it gets closer to 0, because that division gives me very high or very low values.

When I represent it, what will I find? Well, that at 90 degrees I have a vertical asymptote, and then, on the sides, the function either goes to plus infinity or goes to minus infinity. We'll see this later, but how can I know where the function goes?

In this case it would be very simple, I take the calculator, I do the tangent of 89, which is a little to the left of 90, I do the tangent of 89 and you will see that it gives very high values, because the function goes up, towards the infinite.

But if you use the calculator and do the tangent of 91, which is a little bit to the right of that vertical asymptote, you will see that it gives a very low number. So it means that on the right the function is going to minus infinity. Then we will see when we have to represent it.

We continue 180, the same, the tangent of 180 would be the sine of 180 which is 0, between the cosine of 180 which is -1, 0 between -1, if it is 0, then I have the point of coordinates 180, 0. With 270 we have problems again because the sine is -1 but the cosine is 0, -1 between 0, again I have here

plus or minus infinity, which is going to be represented as a vertical asymptote. And then the angle of 360, which is having given a complete turn, and it is like the angle of 0. So it gives 0 divided by 1, which is 0. Then I have the point 360 degrees.

In reality this is also a periodic function, if you notice I have the point, vertical asymptote, point, vertical asymptote, point, and if I were to follow after 360, it would be 360 + 90 = 450, which would also give me vertical asymptote, etc. Then I fill in all these intervals.

How is the tangent function when I represent it? Well, I have already said it a little before, we would have to give values a little to the left of 90 and a little to the right of 90 to see where the function is. A little to the left of 270 and a little to the right of 270. I try with the calculator to see where the function is going. And I already tell you that the tangent function of x at the end has this form.

It is a function that has many discontinuities, cut by these vertical asymptotes, then it comes from the minus infinity, it comes from below, it goes up, it cuts the point zero, well, it cuts the axis of coordinates in the zero, in the 180, in the 360, and then it tends again towards the vertical asymptote, upwards. It is like a function that is always growing. It comes from the minus infinity, it goes through the horizontal axis and it goes towards the infinity.

It is also a periodic function, but much less practical and much less applicable than sine and cosine, because sine and cosine, which were waves, are seen in the whole physics part when you study the simple harmonic movement, like ondulatory exercises, light, waves, etc. But the tangent function, in the end, has less fun.

Well, we have already seen sine, cosine and tangent and we are going to start to see how we can modify and affect these functions. I make a very quick summary of how the sine function was. We put the axes and we give the values that we already have marked there. The sine of 0, which is 0, then I have the point 0, 0. The sine of 90 was 1, point 91. The sine of 180 was 0 again. The sine of 270 was -1.

and the sine of 360 was 0. With these points I can draw, and knowing that it has a wave shape, I can draw this function with its periodic character that is so clearly seen. Now, we had already seen this function in the previous video. Now, what happens if instead of the sine of x I do the function

2 times sine of x. I have a number there that is multiplying, even if I don't put it, 2 times sine of x, which is the double of sine of x. Let's represent it. I use those same values. What happens when x is 0? Sine of 0 is 0 and 2 times 0 is 0. Then here it doesn't matter if I multiply by 2, because since it was 2 times 0, 2 times 0 is still 0. What happens when x is 90 degrees? The sine of 90 was 1,

and now 2x1 is 2. Then instead of the point 91 that was before, I go to the point 92. That point is raised more.

180, the sine of 180 is 0, it was 0, it is still 0, and 2 by 0 is 0. Then this does not vary, I have the point 180, 0. With 270 there is a modification, the sine of 270 is -1, so now multiplied by 2 gives me -2, then I have the point 270, -2. And in 360, the sine of 360 was 0, and 0 by 2 is 0, then it is here. What has happened with this function? That when multiplying it by 2,

All values are multiplied by 2, they are the double, but of course when it is 0, 0 for 2 is the same. However, the other values are stretching, and then what has happened? That the function 2 sine of x is a function as longer, that the ceiling instead of being the values 1 and -1 are the values 2 and -2. It reaches higher and lower. I have achieved, so to speak, to stretch that function.

The opposite would happen if I multiply the function sine of x by 0.5. Why? Because I take these values. When x is 0, sine of 0 is 0, so 0.5 is 0. Now, when x is 90, the sine of 90 was 1, and 1 times 0.5 is 0.5. Then I have the point 90,

0.5, you see it is not as high as before, it is more flat. The sine of 180 was 0 and multiplied by 0.5 is still 0. 270, the sine of 270 was -1 and now 0.5 times -1 is -0.5, then I have the point 270 - 0.5.

And the 0 of 360 was 0, and 0 by 0.5, 0. Then, what has happened if I now add these points? That I have a wave, but flatter. It has less amplitude. It reaches, like a lot, 0.5, and like a little, 0.5.

Then I say goodbye to this video, concluding a very important thing that we have seen, both in the sine function and in the cosine function, if I have a number here that is multiplying, this number is going to modify the amplitude of the wave, ok?

And in fact, with this intention I called "a" to this number, and you will surely find it in physics, in high school, when you study this. This number is responsible for seeing how far the wave will go, as much and as little, instead of 1 and -1, and I go to the next characteristic,

the maximum and minimum values of the function will now be a and -a. Before they were 1 and -1, but if I multiply it by a value, then when multiplying 1 by the value it will give me the value and by -1 the minus value. Then I get that if I multiply that function by 3, then the highest value to which it will reach will be 3 and the lowest, the -3.

And we have already seen it, if the value a is a positive value, it is a large value, then the function will have this shape, it will gain in amplitude, they are like these waves of high frequency, of low wave length.

And if the value A is a low value, then I have a flatter function, a more flat wave, one that has a lot of wave length and little frequency, these are low waves. We won't go into this now, but it's a bit what we've understood, what is the role of a number that is multiplying sine and cosine. In the next video we're going to understand how other parameters that are there in the function act.