Exercises of right triangles in where trigonometric functions must be applied
breast, cosine and tangent, right here right now. on this channel, Math with John. Guys
and girls, how are you? Look at that triangle so pretty I have. It's not a right triangle
but I know, I know the value of one side. And what I want to calculate is the height. Well
So, what am I going to do to calculate the height? Well, what I'm going to do is divide,
divide this triangle, which is not a rectangle in two triangles. Look, on one side we have
this triangle, forty-five degrees, This is the height and I don't know what it's worth.
this side, I'm gonna call this side "x," I'm gonna to call this side "x", this what is here
I'm gonna call it an "x." And on the other hand I have this other right triangle whose angle
it's worth 30 degrees and this piece here, if this is worth 10, this piece that is here
it's gonna be worth 10 minus "x," units: meters. Well, this would be 10 minus "x" and this
would be "h." "h" and "h". The drawing has not very well done, "h" has to have the same
length, but good... Let's see, I have these two right triangles and what can I do?
Apply? What can I apply? Well, look, the trigonometric function that comes to us
as a ring to the finger is the tangent of the angle. Look, the tangent of 45 degrees is going to be the same
on this side split on this other side. And another, on the other hand and worth the redundancy,
the tangent of 30, of thirty degrees, in this case, it's going to be worth this side divided between
this one, ten minus "x." All right. This it means we have a system, we have
a system of equations, two equations with two unknowns. The unknowns are "h and
"x". Let's draw, let's write the equations here: tangent of 45 equal to "h" match
of "x" and tangent of 30 equal to "h" divided between 10 minus "x". Great. Good. What
do we have to do now? Of course, we have to solve this system. And how are we gonna
to solve it? Well, look, I'm going to clear the "h" in this equation, I'm going to clear the "h."
of this equation and I'm going to match, I mean, I'm going to use the equalization method.
I don't need this anymore. I'll get rid of it. If I clear from here
the "h" I'm going to have that "h" equals ... then "x" by tangent of 45 and if we clear here.
the "h" I'm gonna have to 10 minus "x" multiplied. by tangent of 30 because this is "h". And "h"
is equal to this and "h" is equal to this. So I can write that, so that "x" tangent
of 45 is the same as 10 minus "x" multiplied by tangent of 30. Arrived here at this
point it would be very good to remember, without using the calculator, how much tangent is 45
and how much a tangent of 30 is worth. Boys and girls, very interesting to be able to remember how much they're worth
the trigonometric functions sine, cosine and tangent for remarkable angles, for
the most important angles such as angle zero, angle thirty, angle forty and
five, angle sixty and angle ninety. In this case tangent of forty-five
is going to be equal to one and a tangent of thirty. is going to be the same following three games of
three. Well, let's replace these values in our equation. Substituting these values
in our equation we will simply have that "x" is equal to ten minus "x" by root
of three games out of three. Well, we're focused in solving this equation and I'm going to write
this equation up here and I'm gonna operate. I have to clear the "x", I have to clear the "x",
the "x." This is where I'm going to go from here. And operand A little I'm gonna get that "x" equals 10.
root of three match of three minus root of three matches of three "x's". I have here a
"x," here's another "x." If I add both members, if I add both members root of
Three out of three, I'm gonna be able to write. the following: "x" plus root of three party
of three "x" equal to 10 root of three party out of three. And from here I can draw a common factor.
Come on, I take common factor to the "x": "x" that multiply one more root of three match
of three and this equals 10 root of three match out of three. And the last step, guys, and we'll have
achieved "x". "x" is going to be equal to 10 root of three parties of three and all of it divided.
Between... look, this thing here I'm going to operate on. a little bit. Three .... to see, one more root of
three out of three, this can be written as three more root of three match of three.
So what's in here.... I will erase this because I don't need it, I'm gonna erase this.
A little bit... this I erase it like this, like this, like this. I'm gonna leave it, I'm gonna leave
here immersed in these operations. "x" can be simplified as 10 root of three party
of three plus root of three. That's it. This would be units... meters. Look, I could
use the calculator but I'm not gonna do it, I'm not gonna do it. This is enough to
express how much the "x" is worth. And let us remember that what we want is height. It's enough
take this value to either of these two equations, in particular I believe that this me
it turns out, I'm going to find it more comfortable to handle it. All I have to do is to clear the
"x," sorry, the "h" that had and had done it. before. Let us remember that tangent of forty
and five this was 1, this was one and "x" was what's in here. Well, "h" finally
-and this is the end of the exercise- is simply "x", which is this, multiplied by 1, that is,
"h" and "x" have the same value. So "h" shall be equal to 10 root of three divided by
three more root of three and this meters. Guys, girls, red coloring, this exercise
so beautiful is over, this exercise so beautiful is over. I have nothing left
but to say goodbye to you, to advise you that you subscribe to my channel and nothing else.
Come on, see you in the next video, until soon, bye, bye.
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