Trigonometry

This is a 10-lesson curriculum designed for students in the beginning of a trigonometry unit in an Algebra II/Trig course or a Precalculus course. The ideal student has already seen triangle trigonometry (think: SOHCAHTOA), but they have not done much, if anything, with the unit circle. In this two-week unit, they will build the unit circle and come to a deep understanding of the trigonometric functions and their relationships.

From you to eternity and consciously back in an infinite/eternal stand still hypersphere

To show you how unconsciously unconscious you are, I'll just begin with asking rhetorically what you know about your dreams, that constitute the great majority of your mind’s lifetime. Basically nothing …

Then I’ll proceed by showing that you are totally unable to make yourself really understood by anybody: Direct conscious communication without contextual triangulation and observe monopoly cooperation zero.

Now think of the media Putin, Mussolini, Salvini, etc.  and that you do not understand that they are not real characters but only your unconsciously unconscious media projections, of whose reality you know nothing.

Geometry (trigonometry) shows us how to infer a segment by knowing an angle and a side within a circle ⭕️

From this you can project your conscious consciousness on the entire universe by extending the idea that at any radius a sphere also has the same number of points.

This "conscious" conceptual extension can probably be extended to space-time through Lorentz transformations.

Finally - and relatively to Einstein’s happiest thought that everything is “moving” in a (ndr: evolutionary, adaptive, antropologicaMindly) flexible space time with reference to a standstill inertial ALL ON ALL POV - consider that C stands “also” for consciousness:)

PS: To verify you can ConsciousMindly”  "integrate" that your unconscious unconsciousness follows from Gödel's incompleteness, that McLuhan explains your media mirage with the medium = message and that ultra-Darwinism includes "technology" in human evolution, such as any material culture and language.

Da te all'eternità e torna consapevolmente in un'infinita/eterna ipersfera ferma

Per mostrarti quanto sei inconsciamente incosciente, inizierò semplicemente chiedendo retoricamente cosa sai dei tuoi sogni, che costituiscono la grande maggioranza della vita della tua mente.  Praticamente niente…

Poi procederò mostrando che sei totalmente incapace di farti capire davvero da nessuno: comunicazione diretta cosciente, senza triangolazione contestuale e cooperazione dell’osservatore cosciente monopolistica, zero. 

Ora pensa ai mediatici Putin, Mussolini, salvini ecc. e che non capisci che non sono personaggi reali ma solo tue proiezioni mediatiche inconsciamente inconsce, della cui realtà non sai nulla.

La geometria (trigonometria) ci mostra come dedurre un segmento conoscendo un angolo e un lato all'interno di un cerchio ⭕️

Da ciò puoi proiettare la tua coscienza cosciente sull'intero universo estendendo l'idea che in qualsiasi raggio anche una sfera abbia lo stesso numero di punti.

Questa estensione concettuale “consapevole” può essere verosimilMente estesa allo spazio tempo attraverso le trasformazioni di Lorentz.

Infine - e relativamente al pensiero più felice di Einstein che tutto è "in movimento" in uno spazio-tempo (ndr: evolutivo, adattivo, antropologico) flessibile con riferimento a uno stallo inerziale ALL ON ALL POV - si consideri che C sta anche per coscienza:)

PS: per verificare puoi coscienteMente “integrare” che la tua incosciente incoscienza consegue dell'incompletezza di Gödel, che McLuhan spiega il tuo miraggio dei media con il mezzo=messaggio e che l’ultradarwinismo include la “tecnologia” nell’evoluzione umana, come qualsiasi cultura materiale e linguaggio.

It's known that  Shen Kou formula[1 , page 39] can be an approximate method , to define the small angles as a function of  its sine and versine[2] (vers 𝑥 = 1 − cos 𝑥) . In this note by using some numerical methods , we studied Shen Kou formula and  modified it , to be applicable for any  acute angle  .

Trigonometry is a field of study of the long and large angles of a triangle. Based on the results of field observations indicate that trigonometry is quite difficult to be understood by science students in class XI. In addition, the teachers also find it hard to teach the material in a simple way in order to make the student understand it. This is the background of the application development of trigonometry learning system. This application was developed with multimedia-based feature that is accompanied by images, sounds, animation and text, supported with multimedia, trigonometry would be interesting for the students to learn. Applications are developed with Macromedia Flash and action script programming language.

All the articles have been derived by the author Mr H.C. Rajpoot by using HCR's Inverse cosine formula & HCR's Theory of Polygon. These formula are very practical & simple to apply in case of any tetrahedron to calculate the internal (dihedral) angles between the consecutive lateral faces meeting at any of four vertices & the solid angle subtended by it (tetrahedron) at the vertex when the angles between the consecutive edges meeting at the same vertex are known. These are the generalized formula which can also be applied in case of three faces meeting at the vertex of various regular & uniform polyhedrons to calculate the solid angle subtended by polyhedron at its vertex.

Lógica, Números, Funciones, Conjuntos, propiedades de las operaciones entre conjuntos, cuantificadores,  números reales,  expresiones algebraicas, valor absoluto, Ecuaciones, Inecuaciones, Técnicas de conteo,  Teorema del Binomio, Sucesiones,  Trigonometría, Números complejos, Geometría plana, Semejanza y Congruencia, Circunferencia y circulo, Geometría del espacio, prisma, poliedro, área de poliedro, cuerpos de revolución, estadística descriptiva,  medidas de tendencia central y no central,  medidas de dispersión, media, mediana, moda,  probabilidades,  conjunto y probabilidad.

All the important parameters of a small rhombicuboctahedron (an Archimedean solid having 8 congruent equilateral triangular & 18 congruent square faces each of equal edge length) such as normal distances & solid angles subtended by the faces, inner radius, outer radius, mean radius, surface area & volume have been calculated by using HCR's formula for regular polyhedrons. This formula is a generalized dimensional formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can also be used in analysis, designing & modelling of truncated polyhedrons.

Istilah sudut istimewa dalam trigonomerti artinya dalah sudut-sudut yang nilai perbandingan trigonometrinya dapat ditentukan secara eksak. Sudut istimewa merupakan sebuah sudut dengan nilai perbandingan trigonometri yang bisa ditentukan tanpa menggunakan kalkulator. Sudut-sudut istimewa dalam trigonometri adalah 0°, 30°, 45°, 60°, dan 90° dan seterusnya. Untuk mencari nilai sinus, cosinus ataupun tangen pada sudut tertentu kita akan membutuhkan alat bantu hitung untuk dapat menemukan jawaban secara tepat. Namun pada sudut istimewa, tidak membutuhkan alat bantu hitung. Misalnya, pada sudut-sudut istimewa pada kuadran I adalah 0°,

All the articles, related to three externally touching circles, have been derived by using simple geometry & trigonometry to calculate inscribed & circumscribed radii. All the articles (formula) are very practical & simple to apply in case studies & practical applications of three externally touching circles in 2-D Geometry. Although these results are also valid in case of three spheres touching one another externally in 3-D geometry. These formula are also used for calculating any of three radii if rest two are known & the dimensions of the rectangle enclosing thee externally touching circles.  Here is also the derivation of a general formula to compute the length of common chord, angle of intersection & area of intersection of two intersecting circles.

This is a 10-lesson curriculum designed for students in the beginning of a trigonometry unit in an Algebra II/Trig course or a Precalculus course. The ideal student has already seen triangle trigonometry (think: SOHCAHTOA), but they have not done much, if anything, with the unit circle. In this two-week unit, they will build the unit circle and come to a deep understanding of the trigonometric functions and their relationships.

Approximate value of solid angle subtended by an elliptical plane at any point lying on the axis passing through the centre can be calculated by using HCR's Approximation Formula which is applicable on any plane having at least two axes of symmetry & factor of circularity closer to unity for a given location of the observer in the space. It is equally applicable for  rectangle, square, rhombus, ellipse, circle & all other symmetrical planes having at least two axes of symmetry.

In every introductory Calculus course it is proven that:  lim┬(x→0)⁡〖sin⁡x/x=1〗.  This fundamental limit allows to algebraically solve many other trigonometric limits, and also helps to determine the derivatives of the trigonometric functions sin x and cos x, fundamental to find the derivatives of the rest of trigonometric functions. There are different ways to evaluate this limit, being the most widely used the Squeeze Theorem approach, very geometric and clear. But despite the fact that this limit is the key to solve many problems and obtain important results, it always magically appears in classes and textbooks, and nobody explains where this expression comes from or if it is involved or appears in any practical or real life problem, it is just the essential, special, or important result that begins the Trigonometric Limits section. This paper shows that this fundamental limit naturally appears and its value is determined when the Exhaustion Method is used to find the formulas for area and perimeter of the circle

All the articles are related to the reflection of any point about a line in 2-D co-ordinate system and about a line & a plane in 3-D co-ordinate system. Point of reflection about a line or a plane can be easily determined simply by applying the procedures explained or by using formula derived here. These formulas are also useful to determine the foot of perpendicular drawn from a point to a line or a plane in 3-D space. All these derivations are based on the application of simple geometry.

This formula was derived by Mr H.C. Rajpoot by applying his "Theory of Polygon" to calculate all the important parameters of any regular n-polyhedron (out of five platonic solids) such as inner radius, outer radius, mean radius, surface area & volume. This formula is a generalized dimensional formula which can be applied on any existing regular n-polyhedron as it depends on two parameters of any regular polyhedron as the no. of faces & the no. of edges in one face only. It can be used for the analysis, designing & modelling of regular n-polyhedrons.

These tables have been prepared by the author Mr H.C. Rajpoot by using his data tables of the various polyhedra for determining the dihedral angle between any two adjacent faces with a common edge of different uniform polyhedra or polyhedral shells. These are very useful for the construction & preparing the wire-frame models of the uniform polyhedral shells having different regular polygonal faces. A polyhedral shell can be easily constructed/framed by continuously fixing all its adjacent (flat) faces each two as a pair at their common edge at an angle equal to the dihedral angle between them. These tables are very useful in analysis, designing & modeling of various uniform polyhedrons.

The four color map theorem states that no more than four colors are needed to color the regions of any map so that no two adjacent regions have the same color. More specifically, the theorem states that no more than four colors are required to color the vertices of any planar graph where any two adjacent vertices don't have the same color. This paper shows how the theorem arises from a broken correlation between the number of colors and the number of vertices (dots), and, how repairing this broken correlation leads to maps requiring more than four colors. However, if three dimensions are not invoked on a on planer graph, it can be shown that no more than four colors are needed to color it. Premise: All maps, regardless of their shapes and sizes, can be represented by planer dot-and-line diagrams. Such drawings make analysis easier and more universal. Each dot (vertex) represents a country and each line represents a border. The rules are any two dots connected by a line cannot be the same color and lines cannot intersect.

Here is the derivation of a general formula to compute the correct value of the solid angle subtended by any tetrahedron at its vertex when the angles between consecutive lateral edges meeting at that vertex are known because there is no approximation in the formula. This is an analytic & precision formula to compute the correct value of solid angle subtended by a triangle at the origin which is equally applicable in all the cases given the position vectors of all three vertices in 3D coordinate system.

All the standard formula from 'Advanced Geometry' by the author Mr H.C. Rajpoot have been included in this book. These formula are related to the solid geometry dealing with 2-D & 3-D figures in 3-D space & miscellaneous articles in Trigonometry & Geometry. These are useful standard formula to be remembered for case studies & practical applications. Although, all the formula for the plane figures (i.e. planes bounded by the straight lines only) can be derived by using standard formula of right triangle that has been explained in details in "HCR's Theory of Polygon" published with International Journal of Mathematics & Physical Sciences Research in Oct, 2014.
And the analysis of oblique frustum of right circular cone has been explained in his research paper 'HCR's Infinite-series' published with IJMPSR

Teaching Mathematics in the New Normal requires a lot of effort to get students' interest to make them learn independently due to limited time of teacher-student interaction. This study was conducted to determine the effectiveness of Art Integration in teaching Trigonometry to selected Grade 9 online distance learning students of Paliparan National High School, School Year 2020-2021. The study utilized the true design particularly the pre-test-post-test control group design using 60 students chosen randomly from the four Online Distance Learning Modality sections in Grade 9. A standardized 30-item test from DEPED-Division of Dasmarinas was used as the main instrument of the study. Frequency, percentage, mean, standard deviation and t-test for correlated and independent means were used as statistical treatments to answer the objectives of the study. The control group used the existing method or DepEd's Learning Management System while the experimental group used the modules with Art Integration for their asynchronous learning tasks. The computed t-value of 5.633 revealed that the pre-test and post-test scores of the students taught with the use of Art Integration are significantly different. Thus, there is an improvement in the students' performance after being taught using Art Integration. Likewise, the computed t-value of 1.627 on the mean gain of the two groups is not significantly different. Thus, the two groups of students have the same knowledge gained in Trigonometry after being taught. Therefore, the

El presente artículo expone una serie de fórmulas relacionadas con el tema de
triángulos como propuesta para su práctica en el ejercicio de las matemáticas. En
el salón de clases, es una costumbre calcular las áreas y perímetros de diversos
triángulos dados todos o algunos de sus lados, y así, con los diversos teoremas y
reglas de la geometría plana, es posible completar dicha tarea, pero no podemos
decir lo mismo en el sentido opuesto, es decir, desarrollar ejemplos o ejercicios en
donde se proporcione el área y el perímetro, y en base a estos datos, calcular los
lados del triángulo, esta actividad es prácticamente nula en el ámbito académico,
por ende, en este material se ofrecen fórmulas para trabajar con esta alternativa,
como producto del trabajo y la investigación del autor.

No mainstream math professor or math educator I've ever known, could tell the difference between C : 2r and C/2r.  In fact, most people mistakenly believe these are the same thing. The difference is both significant and important. One can't understand the concept of number without understanding this difference.