38. Trigonometry - JulTob/Mathematics GitHub Wiki

📗 Trigonometric Ratios

These “functions” are defined in terms of the sides of a right-angled triangle. We call them O for Opposite a particular angle, A for Adjacent to it, and H for Hypotenuse, the long side. Then sine = O/H, cosine = A/H, and tangent = O/A. An incredible world of relationships comes out of those three simple definitions. Trigonometry was a development of the greatest importance in the progress of mathematics, astronomy and practical arts such as surveying and fortification. The other three functions are easily to be seen to be the reciprocals of the first three. cosec α = H/O = 1/sin α; sec α = H/A = 1/cos α ; cotan α = A/O = 1/tan α.

Razones trigonométricas de un ángulo.


╋╌╌╌╌╌╌╌╌╌╌╌╌╌╌╌╌╌╌╌╌
╋╌╌╌╌╌╌╌╌╌╌╌╱│╌╌╌╌╌╌╌
╋╌╌╌╌╌╌╌╌╌╌╱‿│╌╌╌╌╌╌╌    Sen A=a/c  (Cateto opuesto partido por la hipotenusa)
╋╌╌╌╌╌╌╌╌╌╱ A│╌╌╌╌╌╌╌    Sen A=a/c  (Cateto opuesto partido por la hipotenusa)
╋╌╌╌╌╌╌╌╌╱   │╌╌╌╌╌╌╌    TanA = a/b (Cateto opuesto partido por cateto contiguo)
╋╌╌╌╌╌╌╌╱    │╌╌╌╌╌╌╌         = SenA / CosA (Seno partido por coseno)
╋      ╱     │╌╌╌╌╌╌╌
╋    c╱      │╌╌╌╌╌╌╌
╋    ╱       │b  ╌╌╌╌
╋╌╌╌╱B      C│╌╌╌╌╌╌╌
╋╌╌╱╮_______┌│╌╌╌╌╌╌╌
╋╌      a     ╌╌╌╌╌╌╌
╋━╋━╋━╋━╋━╋━╋━╋━╋━╋━╋

Algebraic Trigs

$Point(𝒙,𝒚)$
Makes an angle with respect to the origin $(𝟎𝑥,𝟎𝑦)$ such as $𝜃$.

\begin{split}

𝒓 = \sqrt{(𝒙-𝟎𝑥)²+(𝒚-𝟎𝑦)²}   \\
  = \sqrt{ 𝑐²+𝑠² }          \\

\end{split}

\begin{split}

𝑐⦨ 𝜃                     \\
𝑠⦬ 𝜃                      \\

\end{split}

\begin{split}

𝚜𝚒𝚗𝜃 = 𝑠∕𝒓           \\
𝚌𝚘𝚜𝜃 = 𝑐∕𝒓            \\
𝚝𝚊𝚗𝜃 = 𝑠∕𝑐             \\
𝚌𝚘𝚝𝜃 = 𝑐∕𝑠              \\
𝚌𝚜𝚌𝜃 = 𝒓∕𝑠               \\
𝚜𝚎𝚌𝜃 = 𝒓∕𝑐                \\

\end{split}

If the point (x,y) of the circle (centered in the origin) satisfies $x^2 + y^2 = 1$ ,then $x=𝑐$ and $y=𝑠$. That is, the equation defines de unit circle.

Las funciones Seno, Coseno, Tangente, Cotangente, Secante y Cosecante son las seis funciones básicas de la trigonometría. De estas además obtenemos sus inversas (o arco). Dándose por entendidas según su relación en círculos y triángulos (venga ya, si no te vas al apartado de Trigonometría, ahora paso), estas son funciones muy interesantes y útiles en el cálculo, dado que tienen propiedades, como la periodicidad, que las hacen muy útiles. Estas son:

  1. Basics

    • tan x = sin x : cos x
    • sec a = √[1 + tan² a ⎤
    • sin x = a / √1+a²⎤

  2. Dominio de definición. Las funciones seno y coseno están definidas en toda la recta real.

    • ∥sin x∥≤１
    • ∥cos x∥≤１

  3. Valores especiales:

    • cos 0 = sen 𝜋⁄𝟸 =1
    • cos 𝜋 = -𝟷

  4. Coseno de una diferencia

    • cos(y-x)= cosy cosx + seny senx

  5. Desigualdad fundamental: Para 0<x<𝜋⁄𝟸

    • 0 < cosx < senx⁄x < 𝟷⁄cosx

  6. Tambien satisface

    • $sen²x + cos²x = 𝟷$

    • $sen (0) = cos( 𝜋⁄𝟸) = sen (𝜋) = 0$

    • Paridad de coseno:

      • $cos(-x) = cos(x)$

    • Imparidad de seno:

      • $sen(-x) = -sen(x)$

    • Correlaciones: ∀x

      • sen(x + 𝜋⁄𝟸) = cos(x)
      • cos(x + 𝜋⁄𝟸) = - sen(x)

    • Periodicidad Trig(x) = Trig(x+2𝜋)

      • sen(x) = sen(x+2𝜋)
      • cos(x) = cos(x+2𝜋)

    • Adición

      • cos(x+y) = cosx cosy - senx seny
      • sen(x+y) = senx cosy + cosx seny
      • sen(a-b) = sena cosb - cosa senb
      • cos(a-b) = cosa cosb + sena senb

    • Diferencia

      • senA - senB = 2 sen( [A-B]⁄2 ) cos( [A+B]⁄2)
      • cosA - cosB = - 2 sen( [A-B]⁄2 ) sen( [A+B]⁄2)

    • Monotonía: En 𝑥∈[0,𝜋⁄𝟸]

      • sen(x) : Estrictamente creciente
      • cos(x) : Estrictamente decreciente

    • Ángulo Doble

      • sen 2𝑥 = 2 sen𝑥 cos𝑥
      • cos 2𝑥 = cos²𝑥 - sen²𝑥 = 1- 2 sen²𝑥

    • Angulo mitad

      • cos²(a/2) = [
1+cos(a)]/2
      • sen²(a/2) = [
1-cos(a)]/2

    • Cuadrados

      • 1+tg²(a)= 1 / cos²(a)
      • cos²a = 
[1 + cos(2a) ]/ 2
      • sen²a =
 1 - cos(2a)
 ]/ 2

    • Producto

      • sen(a+b) + sen(a-b) = 2 sena cosb
      • senA + senB = 2 sen( [A + B]/2 ) cos( [A-B]/2 )
      • cos(a+b) + cos(a-b) = 2 cosa cosb ;
cosA + cosB = 2 cos( [A + B]/2 ) cos( [A-B]/2 )
      • cos(a+b) - cos(a-b) = -2 sena senb ;
cosA - cosB = 2 sen( [A + B]/2 ) sen( [A-B]/2 ) 


    • Cotrigonométrics

      • $sec²θ - tan²θ = 1$
      • $csc²θ - cot²θ = 1$

Angles

Circulo Grados Radians Sin Cos Tan
0 0º 0 0 1 0
1/12 30º π/6 1/2 √3/2 1/√3
1/8 45º π/4 √2/2 √2/2 1
1/6 60º π/3 √3/2 1/2 √3
1/4 90º π/2 1 0 ∞

Right: $⦜$ An angle of $90°$ or $π/2$ radians.
Acute: $⦟$ Sharp, Sharply Pointed. An Angle that is less than a right angle.

Radians

One radian is the distance from the centre to the arc of the circumference along the given arc.

Therefore, this relationship always applies:

𝑠 : Length of the arch
𝑟 : radius
𝜃 : angle in radians
𝑠 ＝ 𝜃·𝑟

In a unit circle, where 𝑟:=1, radians provide the length of the arch too. That's why when omitted radius we assume a unit circle.

In radians, tne complete circumference is defined to be

𝜏 = 2𝜋

Trigonometric Functions

Given a unit circle located on the central axis, we can develop certain trigonometric functions in regard to the point (𝘹,𝘺) in the circle.

sin 𝜃 = 𝘺/𝑟
cos 𝜃 = 𝘹/𝑟
tan 𝜃 = 𝘺/𝘹
csc 𝜃 = 𝑟/𝘺
sec 𝜃 = 𝑟/𝘹
cot 𝜃 = 𝘹/𝘺

These functions have a Period of 𝜏.

Cosine and secant are Even

Sine and Tangent are Odd

cos(a) = cos(–a)
sen(a) = –sen(–a)

Trigonometric Coordinates

We can represent any point in the plane by the distance to the origin and the angle to a given axis (we assume x, generally)

𝘺 = 𝑟 · sin 𝜃 
𝘹 = 𝑟 · cos 𝜃

Trigonometric Identities

By the pitagorean identity

 𝑟² =  𝘹² + 𝘺²

we know that:

 1 =  sin² 𝜃 + cos² 𝜃 
      Sin²θ + Cos²θ = 1

Angulos complementarios

sin(π/2 - θ) = cos θ
cos(π/2 - θ) = sin θ
tan(π/2 - θ) = ctg θ

Angulos Suplementarios

sin(π - θ) =  sin θ
cos(π - θ) = -cos θ
tan(π - θ) = -tan θ

Ángulo Recto

sin(π/2 + θ) = cos θ
cos(π/2 + θ) = -sin θ
tan(π/2 + θ) = -ctg θ

Ángulo Llano

sin(π + θ) = -sen θ
cos(π + θ) = -cos θ
tan(π + θ) = tan θ

Angulo doble

sin(2θ) = 2·sinθ·cosθ
cos(2θ) = cos²θ - sin²θ
tan(2θ) = 2·tanθ
          / 1 tan²θ

Angulo Mitad

sin(θ/2) = 土√(1-cosθ) /2
cos(θ/2) = 土√(1+cosθ) /2
tan(θ/2) = 土√(1-cosθ) /√(1+cosθ)

Trigonometry and Complex Numbers

Trigonometry and Complex Numbers work very nicely together. It is important to remember Euler's Identiy.

e( j θ) = cos θ + j  sin θ

Other Identities


Cos²θ - Sin²θ =  cos 2θ

Cos²θ =  (1+ cos 2θ )⁄₂

cos(a+b) = cos a · cos b - sin a · sin b

Cos² (a) = (1 + cos (2 a) ) / 2
Sin² (a) = (1 - cos (2 a) ) / 2

Integrales Trigonométricas

⌠𝑎
⎮ cos𝑥 𝖽𝑥 = cos𝑎
⌡0

⌠𝑎
⎮ sen𝑥 𝖽𝑥 = 1 - cos𝑎
⌡0

Con a expresado en forma de radianes.Siempre.

Trigonometry ⊿⊾∠⩗⩘⪦⪧⦛⦜⦝⦞⦟⦠⦡⦢⦣⦦⦧⦨⦩⦪⦫⦬⦭⦮⦯⧡ △⍋⍙⎖⏅⏃▶︎▷▹◃▿▵△◭◮◬◢◣◤◥◸◹◺◿∆∇⊾⊴⊳⊲⊵⊿⋈⟓⟔⦛⦝⦞⦠⦡⦵⦶⧎⧋⧌⧍⧊⧖⧨⧩╱╲°º۞⊙⊲⊳⊾⊿⌔⍋⍙◌○●◐◑◒◓◴◷◶◵⟁⟑⟳⟲⦛⦝⦞⦠⦡⦢⦥⦤⦣⦵⦶⬤🌐📐📏📶◠◡◜◝◞◟◜◕◴◵◶◕◷◔◗◐◑◒◓◖◌❍◯⥀↺↻⏦⌓⌔⌒⌖⎊⌽⎋◢☸︎⌜⌝⌞⌟⎖〇⤽⤼⤺⤻⟲↺↻〜⦈⦇