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jefflownsbury

*Follow*

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jefflownsbury

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I want to understand this connection better.

Although it sounds the most involved, `arctan`

answers the simplest trigonometric question you could ask. “The full moon subtends seven degrees of my vision; so how big is the moon?” The `arctan`

(with radius = distance from Earth to moon) is the answer.

But what does that simple operation have to do with:

- a continuous sum of
- an inversion of
- one more than
- some continuous range of numbers
- times themselves?

That’s confusing.

Four years later…

This comment shows how to get the answer. (Pair with this on the derivative of logarithms.) A consequence of the chain rule (or perhaps another way to state it!) is that the derivative of ƒ⁻¹ at a point p is `flip( derivative( ƒ )) evaluated at ƒ⁻¹(p)`

, presupposing that all of the maps fit together right.

`arctan`

is defined, for simplicity, on a circle of unit radius. (This just means if you’re looking at the moon then use units of “one moon-distance”.) It takes as an argument a ratio of sides and returns an angle `θ`

. Since `derivative ( tan ) = derivative (sin⁄cos) = 1⁄cos² = sec × sec`

(reasoned with calc 101), by combining that `derivative ( tan )`

with JavaMan’s perspective on the derivatives of inverse-functions we can argue that

Java Man’s idea gets us to look at the triangle

which, since it’s constrained to a circle by the equivalence-classing of triangles to be just the ones with a certain angle (see fibration), limits us to just one free parameter

`R`

(a ratio of opposite `O`

to adjacent `A`

side lengths of an equivalence-class of triangles). After following Java Man’s logic to see why `derivative( tan ) = sec²`

implies `derivative( inverse of tan ) = cos² [at the angle which is implied by tan R]`

, we’re left knowing that `A`

=the adjacent side implied by the `cos R = cos O⁄A`

of the original `O⁄A`

ratio we were given as the natural input space (ratios) which the `tan`

function accepts. This isn’t enough because the answer needs to be in `O⁄A`

terms to match the input. We have to do some Pythagorean jiu jitsu involving `A = A⁄1`

and `1=1²=A²+O²`

to get the answer into an `O⁄A`

form (since that was the information we were given). Using the `1=A²+O²`

is using the natural delimitation of the inscribing circle to make the two `A`

and `O`

move together the way they should on the circle, by the way. The algebraic jiu jitsu then yields `A = flip( 1+R² )`

, now using the proper input `R=O⁄A`

.
The point of all this mangling was merely to match up cosine’s output with tangent’s input. Sheesh with all the symbols!

But that’s merely deriving the correct answer algebraically, with a bit of “why” from the comment’s perspective on inverse functions generally. What about my original question? Why does this sequence of mappings, if iterated, subtend the moon?

isomorphismes

Love everything about math but trig #TrigSucks #Sin #Cos #Cosec #Sec #Cot #ArcSin #ArcCos #ArcTan #AndLotsOfAnnoyingIdentities #Sigh

blackabaya

It's very sad

That I know the words to songs in languages I can’t speak but I can’t seem to grasp differential equations or integrations and stuff..Ok maybe I’m just too lazy to grasp the concept of integrating **arctan x **and really who cares(ok I do exam week but that’s another story) all I’m saying is I know the words to this song and this gem of a song Oh how about this one..and yes I know there is a common thread throughout all of them *hint* foreign lesbians(I blame tumblr) but still I should care more about my education or better still stop using awesome songs as soundtracks for sad lesbian moments it’s too distracting.

theuniverseisafinickylilbastard

Su AppWall ospitiamo OneCalc, una calcolatrice fantastica in Material Design semplice da usare e con mille funzioni utilissime come:

● Aritmetica di base

● Integrali definiti

● Funzioni trigonometriche(sin, cos, tan, arcsin, arccos, arctan)

● Numeri complessi

● Calcolo matriciale

● Conversione tra unità di misura

● Grafici di equazioni

E per di più è anche gratis!

Un app davvero da non perdere.

promezio

computers suck, programming sucks, i hate fractions and the arctan function

rukokarasu