cosine

cercando di dimenticare il fatto che i miei se ne stanno andando fuori di casa e io non ho nessuno con cui bombare indisturbata sul divano, non so comunque cosa fare. starmene in camera a guardare film su popcorntime mi sembra uno spreco di tempo :(

anonymous asked:

Ti hanno mai detto che potresti soffrire di disformobico? o come si dice ( scusa sono dislessica)

Dismorfofobia. In passato sicuramente sì, e mi è stato diagnosticato insieme a tante altre simpatiche cosine, ma pesavo 40 chili bagnata e mi vedevo davvero chiattissima e non riuscivo assolutamente a rendermi conto che in realtà non era così; mi pizzicavo le ossa ed io veramente ci vedevo soltanto grasso, facevo la doccia con la luce spenta ed evitavo gli specchi. Adesso non è così grave, so di non essere grassa, ma volersi bene è lo stesso una battaglia sfiancante e quotidiana.

The familiar trigonometric functions can be geometrically derived from a circle.

But what if, instead of the circle, we used a regular polygon?

In this animation, we see what the “polygonal sine” looks like for the square and the hexagon. The polygon is such that the inscribed circle has radius 1.

We’ll keep using the angle from the x-axis as the function’s input, instead of the distance along the shape’s boundary. (These are only the same value in the case of a unit circle!) This is why the square does not trace a straight diagonal line, as you might expect, but a segment of the tangent function. In other words, the speed of the dot around the polygon is not constant anymore, but the angle the dot makes changes at a constant rate.

Since these polygons are not perfectly symmetrical like the circle, the function will depend on the orientation of the polygon.

More on this subject and derivations of the functions can be found in this other post

Now you can also listen to what these waves sound like.

This technique is general for any polar curve. Here’s a heart’s sine function, for instance

The sine and cosine functions for the circle, as every student should see them.

(Edit: the animation is also available, without watermark, at higher resolution and slower frame rate at Wikimedia Commons.)

HAPPY PI DAY! To celebrate, here’s this long-due animation of the usual trigonometric functions, sine and cosine, geometrically defined in terms of the unit circle.

I know this animation is a bit of   the same   as several   others   of my previous   animations  , but this is THE version that I should have done ages ago, if not done first of all.

This is what the sine and cosine functions, the ones you are taught, really are in terms of the unit circle.

First, we have the unit circle (with radius = 1) in green, placed at the origin at the bottom right.

In the middle of this circle, in yellow, is represented the angle theta (θ), that we’re going to plug in our trigonometric functions. This angle is the amount of counter-clockwise rotation around the circle starting from the right, on the x-axis, as you can see. An exact copy of this little angle is shown at the top right, visually helping us define what θ is.

At this angle, and starting at the origin, we trace a (faint) green line outwards. This line intersects the unit circle at a single point, which is the green point you see spinning around at a constant rate as the angle θ changes, also at a constant rate.

Now, we take the vertical position of this point and project it straight (along the faint red line) onto the graph on the left of the circle. This gets us the red point. The y-coordinate of this red point (the same as the y-coordinate of the green point) is the value of the sine function evaluated at the angle θ, that is:

   y coordinate of green point = sin θ

As the angle θ changes, we can see the red point moves up and down, tracing the red graph. This is the graph for the sine function. The faint vertical lines you see passing to the left are marking every quadrant along the circle, that is, at every angle of 90° or π/2 radians. Notice how the sine curve goes from 1, to zero, to -1, then back to zero, at exactly these lines. This is reflecting the fact sin(0) = 0, sin(π/2) =1, sin(π) = 0 and sin(3π/ 2) = -1

Now, we do a similar thing with the x-coordinate of the green point. However, since the x-coordinate is tilted from the usual way we plot graphs (where y = f(x), with y vertical and x horizontal), we have to “untilt” it in order to repeat the process above in the same orientation. This was represented by that “bend” you see on the top right.

So, the green point is projected upwards (along the faint blue line) and this “bent” projection ends up in the top graph’s rightmost edge, at the blue point. The y-coordinate of this blue point (which, as you can see due to our “bend”, is the same as the x-coordinate of the green point) is the value of the cosine function evaluated at the angle θ, that is:

   x coordinate of green point = cos θ

The blue curve traced by this point, as it moves up and down with changing θ, is the the graph of the cosine function. Notice again how it behaves at it crosses every quadrant, reflecting the fact cos(0) = 1, cos(π/2) = 0, cos(π) = -1 and cos(3π/2) = 0.

And there you go. That’s all there is to it. That’s what sine and cosine are. Simple, huh?

Now, while the concept itself is pretty simple, a lot of people get confused about what the sine and cosine functions actually represent, because visualizations such as this are not presented to them when they are first taught trigonometry.

A lot of teachers, and plenty of school books, fail to mention any of this in detail, as I tried to do here, instead throwing a bunch of formulas in front of students. But the geometric intuition, as presented here, is much simpler to grasp, much more useful in general, and will stick to you for life once you get it. The formulas and important values for sine and cosine don’t need to be memorized anymore, because now you should understand what these values should be, given the underlying logic of things. And that’s what math is all about: making sense of things so they are plainly evident to anyone.

In my most popular post to date (over 360 thousand notes as of now, holy crap!), I saw a lot of people commenting that seeing the top graph, which is the sine function for the circle, made all that trigonometry stuff click.

I was baffled. People were angry that no teacher has ever showed anything like that to them before. That’s crazy! At this age where computers are everywhere, this sort of thing should be in every classroom, and be seen by every student.

So, in order to do justice to the unit circle and these immensely important trigonometric functions, and in order to fill an obvious pedagogical hole in math classrooms and textbooks everywhere, I decided to finally make this animation. No fancy or crazy alternative takes on the sine and cosine this time, just the good ol’ pair of trigonometric functions we all should understand and love.

Happy Pi Day, everyone!

ogni volta che conosco un ragazzo e gli dico che il mio genere di danza (ma non solo danza vbb) è l'hip hop posso stare sicura che il giorno dopo il suddetto ragazzo inizierà a fare ricerche sfrenate nel genere e che la volta successiva che usciremo in macchina partirà casualmente un pezzo di tupac o notorious big e questa cosa mi fa proprio tenerezza (anche un po’ pietà) perché non dovete farvi piacere la musica che piace a me per fare colpo se poi siete delle persone dimmerda.

Pi’s parametric coordinate functions

This is the second of three animations I’ll be posting today (here’s the first). Be sure to check them out later if you miss them!

The polygonal trigonometric functions I described earlier were based on the interior angle, instead of the length along the polygon’s border.

This simplified things a lot, and created some interesting uses for the functions. However, since I could only have one value of radius for each angle (they were based on polar equations), I could not draw arbitrary shapes with a continuous line based on the [0,2π] interval.

The solution is to extend the idea to general closed curves, by using the position along the curve to define the sine and cosine analogues. In other words, we want “path trigonometric functions” for which the input parameter is the position along the path, and whose periods are the curve’s total arc-length.

But the concept of “sine” and “cosine”, as well as “trigonometric”, completely lose their meaning at this point. It has nothing to do with triangles or angles.

We’re now dealing with the functions x(s) (in blue) and y(s) (in red) that together describe the curve, by being used in the parametric equation r(s) = ( x(s) , y(s) ), where r(s) is a vector function and s is the arc-length. This is very standard stuff, so it isn’t incredibly exciting anymore.

Notice that if the green curve was a unit circle, the functions would become the usual sine and cosine.

But we do get to see what these functions look like and what they are doing. So here’s the coordinate functions for the arc-length parametrization of a pi curve!

Happy Pi day!

Here’s a relaxing Sin(xy) curve!

Easy Mathematica code if anyone’s interested:

Manipulate[
 Plot3D[Sin[
   x y + \[Phi]], {x, -\[Pi]/2, \[Pi]/2}, {y, -\[Pi]/2, \[Pi]/2},
  Boxed -> False, Axes -> False, Mesh -> False,
  ColorFunction -> ColorData[“BeachColors”], Background -> LightBlue,
  PlotRange -> {{-\[Pi]/2, \[Pi]/2}, {-\[Pi]/2, \[Pi]/2}}], {\[Phi],
  0, 6 \[Pi], \[Pi]/8}]

In a previous post, I showed how to geometrically construct a sine-like function for a regular polygon.

I also pointed out how the shape of the function’s graph depends on the orientation of the polygon, since it isn’t perfectly symmetric like the circle.

This animation illustrates how the polygonal sine (dark curve) and polygonal cosines (clear curve) change as the generating polygon rotates.

Derivation

First of all, it is important to point out these functions are not based on the perimeter of the shape, like it is for the unit circle. We’re still sticking to the interior angle here. If we used the perimeter as a substitute for the angle we would just get a deformed linear spline of the sine function, which is rather useless and boring.

In order to find these functions for an arbitrary polygon, we first need to write the polygon in polar form. That is, we want the radius for a given angle. In a circle, this is a constant value.

A general “Polar Polygon” function is:

PPn(x) = sec((2/n)·arcsin(sin((n/2)·x)))

Where n is the number of sides of the polygon. If n is not an integer, the curve is not closed.

Armed with this function, we can quickly find the polygonal sine and polygonal cosine:

Psinn(x) = PPn(x)·sin(x)
Pcosn(x) = PPn(x)·cos(x)

As n grows, the functions approximate the circular ones, as expected. To rotate the polygon, just add an angle offset to the x in PPn.

This technique is general for any polar curve. Here’s a heart’s sine function, for instance

So, what is it good for?

I’ve used this several times when I wanted some smooth interpolation between a circle and a polygon, in such a way that the endpoints of the interpolation are a perfect circle and a perfect, pointy polygon. It’s useful in parametric surfaces, such as in this old avatar of mine:

Now you can also listen to what these waves sound like