# arctan

4

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:

1. continuous sum of
2. an inversion of
3. one more than
4. some continuous range of numbers
5. 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?

I just wrote an arctan() and arctan2() implementation

because the flipping language didn’t have it defined serverside, only clientside.

Now?  Now I can do pixel-perfect aiming and fancy trig/calculus.  Yay!

Trig skills++;  // <3

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

Yomar Gonzalez ( Androidfast )

One Calculator FULL v2.4.1 build 76
Requisitos: 4.0 y hasta
Descripción: Una Calculator es una calculadora científica con todas las funciones con una interfaz simple y limpia, perfectamente integrado con Google Diseño de Materiales.

Descripción
Uno Calculator es una calculadora científica con todas las funciones con una interfaz simple y limpia, perfectamente integrado con Google Diseño de Materiales. El suministro de las diferentes modalidades de cálculo, Una calculadora es adecuado para la solución de cálculo común fácil y cálculos matemáticos más complejos.
Una de las características de la calculadora:
● Aritmética Básica
● Integrales definidas
● Trigonometría (sin, cos, tan, arcsin, arccos, arctan)
● Los números complejos
Cálculos ● RPN (Reverse notación / notación postfix polaco)
● cálculo Matrix
● Medición de conversión de unidades
● Los gráficos de X, Y ecuaciones
● La conversión entre decimal, hexadecimal, octal y binario
● cronología de las operaciones Intreactive
● Soporte para tabletas y smartphones
Ayúdame a traducir la aplicación: _http: //ackuna.com/translate-/one-calculator-3/

Qué hay de nuevo
- errores menores fijo
- Corregido un error con RPN y comas
v2.4.1
- Un error introducido con la última actualización se ha fijado
- Nueva traducción ruso, gracias a Алексей Кутузов
v2.4
- Ahora corchetes cambiar de color cuando se selecciona, indicando si han sido cerrados o no
- La aplicación puede ahora cerrar soportes de forma automática (para permitir ir a: Ajustes -> Calculadora -> Automáticamente cerrar paréntesis)
- Ahora es posible para evitar operaciones anteriores se guarden
- Añade más sombras en las versiones pre Lollipop de Android

Más Información: