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If you've ever been through... junior high?
you probably are familiar with these two
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trigonometric functions and their respective
plots, creators of eternal stress and frustration
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in students around the world; and it's quite likely that you've solved countless
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problems involving them without really
having an idea of what was going on.
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Well, in this video we are going to try to understand
and comprehend what is the sine and cosine of an angle,
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seeing what it would be like if you discovered
these functions.
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Imagine that you have several right triangles,
that is, they have an angle of 90 degrees
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(also called right angle... right-triangle),
and that have the same proportions,
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that is, if you turn them over, rotate, and resize them,
they should all look identical.
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It's a boring day, so you decide to calculate the result
of dividing the shortest side of each triangle
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by the longest, because why the [bleep]
not?
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You measure the sides with your ruler, do the math,
and quickly discover that every one gives ...
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basically the same ...
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–Will this be quirk of this kind of triangle?– you ask yourself,
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so without changing the size
on the largest side, you modify each triangle slightly,
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while maintaining proportionality and "rightness",
and you repeat the calculations.
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They keep giving all same!
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You do it more times with similar results,
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until what you have are more lines than triangles.
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You're kinda of a nerd, so you take out graphing paper and get ready to plot the results,
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but you run into a problem:
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You know that varying
the shape of the triangle, you get different values,
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but you notice that these don't depend
of its size, the scale doesn't matter
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the ratio between the shortest and
the largest side is always the same.
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You deduce that what you are
changing is the size of this angle,
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and that the values you got depend on it. By changing the angle every 15 degrees, you get
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this graph, and joining the points, it looks like this.
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Well, you just discovered the essence
of the sine of an angle. The division, the ratio,
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between the opposite side of a right triangle and its hypotenuse given a certain angle.
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Okay, if you didn't understand the words I just said, here goes a bit of terminology.
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In a right triangle always, no matter
its shape, we're gonna have a bigger side
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than the other two, which we call hypotenuse,
and it's opposite to the 90 degrees angle, while
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the other two are called just sides (sometimes legs) and
they're named depending on the angle with which
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we are working, for example: If the angle
that interests us (which we usually call
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alpha if we know its value, and theta if not)
is here, this would be the opposite leg,
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and this would be the adjacent one, which means
that is right next to the angle; but if we work
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with this other angle, the names are reversed
and this would be the adjacent one and this one the opposite.
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Anyway, a common way of teaching (and
remembering) trigonometry is with the unit
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circle, which consists of a circle
of radius 1 located in the center of the Cartesian plane.
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We're move this radius by varying the angle it makes with the positive part of the
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x axis in the opposite direction to the hands
of the clock (also called counterclockwise)
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and we are going to form a right triangle
in such a way that the radius is the hypotenuse
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and the right angle is always attached to
x axis, that is, we're not going to do this.
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Recall that the sine is the result of
dividing the opposite leg between the hypotenuse,
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but in this case, the hypotenuse is 1, and
any number divided by 1 is...
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the same number, so the sine is simplified
to be solely the length of the vertical part
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of the triangle, the opposite leg.
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Okay, you probably know that a circle
it is divided into 360 degrees, but this is mathematics,
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so let's use radians like actual mathematicians.
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To know how much a radian is, we take
the radius of the circle (hence the name),
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and without stretching or compressing it, we place it
gently on the circumference. The angle
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of the arc that was formed is a radian. If you know
the formula of the perimeter of a circle,
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you will know that 2π times the radius is needed
to complete the circumference, so
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in a whole circle there is an angle of 2π radians,
in a half circle there are π radians, in a quarter
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But anyway, getting off this tangent,
let's observe and graph (no graphing paper no more)
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how does the value of sine changes with the
angle, which if we remember, is equal to the value
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Notice that from the angle π (180 degrees),
the sine becomes negative because now,
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we are measuring downwards.
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But in maths, we don't just finish at 2π radians, there are as many angles as there are numbers,
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so we can keep going around to
infinity, and even go in reverse to
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Get the famous sine graph.
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If I explained myself well, you understood what is
sine of an angle (if not, skip back the video,
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and if that fails, leave a comment),
so concept of cosine should be relatively
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In this case, we're gonna get the ratio between the adjacent leg and the hypotenuse.
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In the unit circle, it represents the horizontal part of the triangle, the distance of it
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to the y-axis, and its graph is made like this.
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This is pretty much the basics, and I really
hope that you leave this video understanding
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what is the sine and the cosine, since I have noticed
that several students, including at the university level,
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do not understand what these trigonometric functions are at a fundamental level.
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If this video helped you and you want me to go deeper
on the subject (or talk about something else, mathematics
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or not), please let me know in the comments.
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If you want to support the creation of more videos like this one, you can help me through Patreon,
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where you can donate the amount you want
in exchange for perks each time I upload a new video.
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Obviously, you do not have to do it,
and these videos will be free forever,
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but if you want, and you can, well ... please.
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Thank you very much for watching this far, now ...
