Trigonometric Method Research Articles

Generalizing the Equal Incircles Theorem: Insights from Sangaku Problems

Sangaku problems are traditional Japanese geometrical puzzles, often displayed in religious temples, that have intrigued mathematicians for centuries. This study aims to generalize the Equal Incircles Theorem, extending Angela Drei’s proof to N-circles, by applying the trigonometric method alongside foundational mathematical tools, including mathematical induction, Heron’s formula, and the telescoping product. A generalized equation for N circles based on the Equal Incircles Theorem is derived through explicit mathematical formulation and characterization. The findings deepen our understanding of geometric relationships, highlight the historical significance of Sangaku problems, and offer potential advancements for future engineering applications, mathematics education, and research in mathematical history.

Comparative Study of Alternative Methods for Measuring Leg Length Discrepancy after Robot-Assisted Total Hip Arthroplasty.

Our study addresses the lack of consensus on measuring leg length discrepancy (LLD) after total hip arthroplasty (THA). We will assess the inter-observer variability and correlation between the five most commonly used LLD methods and investigate the use of trigonometric principles in overcoming the limitations of current techniques. LLD was measured on postoperative AP pelvic radiographs using five conventional methods. CT images created a 3D computer model of the pelvis and femur. The resulting models were projected onto a 2D, used to measure LLD by the five methods. The measurements were evaluated via Taguchi analysis, a statistical method identifying the process's most influential factors. The approach was used to assess the new trigonometric method. Conventional methods demonstrated poor correlation. Methods referenced to the centers of the femoral heads were insensitive to LLD originating outside the acetabular cup. Methods referencing either the inter-ischial line or the inter-obturator foramina to the lesser trochanter were sensitive to acetabular and femoral components. Trigonometry-based measurements showed a higher correlation. Our results underscore clinicians' need to specify the methods used to assess LLD. Applying trigonometric principles was shown to be accurate and reliable, but it was contingent on proper radiographic alignment.

Spatiotemporal analysis using deep learning and fuzzy inference for evaluating broiler activities

Observing poultry activity is crucial for assessing their health status; however, the inspection process is often time-consuming and labor-intensive, particularly in cases involving large numbers of chickens. Inexperienced breeders may also misjudge their activity levels, potentially missing opportunities for prevention and treatment. This study integrates traditional video surveillance with an advanced monitoring system to identify various broiler behaviors in a breeding environment. A two-stage deep learning approach is employed: in the first stage, the broilers are detected, and in the second stage, five key body points (head, abdomen, two legs, and tail) are identified. A skeleton-based model is then developed centered around the abdomen, with six angles calculated using trigonometric methods. These angles are analyzed by a long short-term memory network to estimate behaviors such as “Standing”, “Walking”, “Resting”, “Eating”, “Preening”, and “Flapping”, selecting the behavior with the highest probability. Dual-layer fuzzy logic inference systems were used to evaluate the proportion of time broilers spent in static versus dynamic states, providing a robust determination of their activity levels. Validated in a mixed-sex breeding environment, the proposed system achieved accuracies of at least 85.2% for identifying broiler type, 79.2% for identifying body parts, and 50.8% for identifying behaviors. The activity level evaluation results were consistent with those conducted by experienced poultry experts.

Stress Characteristics and Mechanical Behavior of Rock Masses with an Opening under Complex Deep Underground Stress Conditions

In this study, the impact of principal stress states on the stress characteristics and initial failure of the rock mass surrounding a three-center arch opening was investigated using complex variable function methods and Discrete Element Method (DEM) numerical modeling. First, the mapping function of the opening was determined using the trigonometric interpolation method, and the influence of the number of terms in the mapping function on its accuracy was revealed. Based on this, the far-field stress state of the underground rock mass was characterized by the ratio of the minimum to maximum principal stress (λ) and the angle (β) between the principal stress and the vertical direction. This stress state was then converted into normal and shear stresses. Using complex variable function theory, the stress characteristics at the boundary of the opening under different stress states were analyzed. Finally, DEM numerical modeling was employed to study the initial failure characteristics at the boundary of the opening and its relationship with the stress distribution. The results indicate that the lateral pressure coefficient significantly affects the stability of the opening by influencing stress concentration around the surrounding rock. Low lateral pressure coefficients lead to tensile stress concentration at the boundary perpendicular to the maximum principal stress. As the coefficient increases, tensile stress decreases, and compressive stress areas expand. While the principal stress direction has a minor effect on stress concentration, it notably impacts stress distribution at the boundary. When λ < 1.0 and β = 45°, stress distribution asymmetry is most pronounced, with the highest compressive stress. The early failure distribution aligns with stress concentration areas, validating the use of stress analysis in predicting opening stability and failure characteristics.

Study on train safety control of high-speed railway bridge under the action of near-fault earthquake

In order to study the effect of the velocity pulse on the dynamic response of the train-bridge system of the high-speed railway simple supported beam bridge, the velocity pulse is simulated by the trigonometric function method and superimposed with the far-field earthquake without pulse to synthesize the pulse with different pulse types, pulse periods and pulse peaks. A 10×\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ imes$$\\end{document}32m typical high-speed railway simple supported beam bridge is considered an case illustrating study. Then, the dynamic response of train-track-bridge coupling system is calculated by train-track-bridge seismic analysis software TTBSAS. Afterwards, the influence of pulse near-field earthquakes parameters and vertical components on dynamic response of train-bridge system and the safety of the train on the bridge are discussed in detail.The new derailment evaluation index is adopted to evaluate driving safety under earthquakes. The train safety control of simply supported beam bridge under the action of near-field earthquake is studied. The results show that the impact of pulse ground motion on the dynamic response of the train-track-bridge coupling system is significantly higher than that of no-pulse ground motion, especially the impact on the bridge and rail subsystem is more significant than that of train subsystem. Under the excitation of ground motion intensity of 0.05g∼\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim$$\\end{document}0.15g, the safe speed threshold of pulse near-field ground motion is smaller than that of far-field ground motion. When the ground motion intensity is 0.20g∼\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim$$\\end{document}0.30g, the safe speed threshold of pulse near-field ground motion and far field ground motion is 200km/h. So, the pulse near-field earthquake poses a greater threat to the safety of the train on the bridge than the far-field earthquake. Therefore, the influence of pulse near-field earthquakes should be considered in the seismic design. The research results of this paper can provide theoretitcal support for the design of a high-speed railway bridge in the near-field area.

POLYNOMIAL SOLUTIONS FOR THE KOLMOGOROV–WIENER PREDICTION OF MODELED SMOOTHED HEAVY-TAIL PROCESS

Nowadays telecommunication traffic in systems with data packet transfer is considered as a heavy-tail random process. In a couple of rather simple models traffic is considered to be stationary one. In our recent papers we generated modeled heavy-tail data, which is based on the smoothing of the fractional Gaussian noise. In particular, the applicability if the continuous Kolmogorov–Wiener filter to the prediction of such data was investigated, the corresponding Wiener–Hopf integral equation was solved on the basis of the truncated Walsh function expansion. However, a question occurs – may another truncated orthogonal function expansion be applied to the problem under consideration? So, the corresponding investigation may be an actual question. In our recent papers we investigated theoretical fundamentals of the Kolmogorov–Wiener filter construction for different models, in particular, on the basis of the truncated polynomial expansion method and on the basis of the truncated trigonometric Fourier series expansion method. In this paper we restrict ourselves to the investigation of the applicability of the truncated polynomial expansion method to the problem under consideration, the corresponding method is based on the Сhebyshev polynomials of the first kind. The applicability of another polynomial or trigonometric expansions to the problem under consideration may be discussed in other papers. The aim of the work is to investigate the applicability of the Galerkin method based on the Chebyshev polynomials of the first kind to the Kolmogorov–Wiener prediction of smoothed heavy-tail data. The methodology consists in the solving of the Wiener–Hopf integral equation on the basis of the truncated polynomial expansion method which is based on the Chebyshev polynomials of the first kind. The scientific novelty consists in the proof of the fact that the Galerkin method based on the Chebyshev polynomials of the first kind may be applied to the Kolmogorov–Wiener prediction of smoothed heavy-tail data. The conclusions are as follows. The truncated polynomial expansion method based on the Chebyshev polynomials of the first kind may give reliable results in the framework of the Kolmogorov–Wiener prediction of smoothed heavy-tail data.

Solve third order Lane–Emden–Fowler equation by Adomian decomposition method and quartic trigonometric B-spline method

In this work, we study the Lane–Emden–Fowler equation which has a wide range of applications in astrophysics, classical and quantum mechanics. Firstly, we convert this nonlinear differential equation into an equivalent integral equation. We employ the powerful Adomian decomposition method to derive an analytical solution that will also lead to numerical solutions. Moreover, we exhibit two theorems to ensure the uniqueness of the solution of the problem and to confirm the convergence of the proposed scheme. In addition, we use quartic trigonometric B-spline as a powerful numerical method to achieve numerical solutions. Finally we present a comparison between the proposed methods to show their accuracy performance. Proper graphs are also provided to illustrate the obtained solutions.

Stochastic Vibration-Driven Fatigue Analysis for Steel Structure Bridges Considering Track Irregularity

This study proposes a method for assessing the effect of stochastic vibrations resulting from track irregularities which are rarely studied but significant on bridge fatigue. First, the pseudo-excitation method is utilized to determine the power spectrum density of stress. Subsequently, the trigonometric series method is employed to acquire the stress time history set. Then, a traditional fatigue life calculation method is employed for analysis. By utilizing a post-sampling method, the need for costly repetitive solutions of the vehicle-bridge coupling system is eliminated. Numerical simulations are conducted on a steel truss bridge, revealing the non-negligible effect of track irregularity on bridge fatigue performance. The study investigates the underlying mechanism, highlighting the pronounced influence of long-wave track irregularity. Moreover, the research explores the correlation between track smoothness and bridge fatigue performance, demonstrating that a decrease in track smoothness leads to an increase in fatigue loss. These findings emphasize the necessity of considering stochastic vibrations when evaluating bridge fatigue, providing valuable insights for bridge design and maintenance.

Measurement of Postoperative Apical Vertebral Rotation Using Radiographic Images in Adolescent Idiopathic Scoliosis-Quantitative Evaluation Using Inverse Trigonometric Functions.

Conventional methods for analyzing vertebral rotation are limited to postoperative patients who underwent posterior fusion. A previous methodology calculated vertebral rotation using inverse trigonometric functions based on the length of the pedicle screw (PS). Accordingly, this study evaluates rotational deformity in patients with postoperative adolescent idiopathic scoliosis (AIS) using inverse trigonometric functions. This retrospective single-center study includes patients with AIS who underwent posterior fusion surgery. Postoperative radiography and computed tomography (CT) scans were retrospectively evaluated. The magnification ratio was calculated using the rod diameter (δ=lateral/frontal rod diameter), and the visible screw lengths were measured using radiographs. The rotation angle was calculated using the apex of the main curve and the lower instrumented vertebra (LIV) (rotation angle=tan-1 (lateral PS length/(δ×frontal PS length))) immediately following the surgery as well as two and five years postoperatively. The correlation between the direct CT measurement and postoperative rotation angle progression was investigated. The crankshaft phenomenon (CSP) and distal adding-on (DAO) were evaluated as postoperative deformities. CSP was defined as a 5° increase in rotation angle. Seventy-eight patients (age: 15.3±2.0 years, eight boys and seventy girls) were included. The rotation angle was strongly correlated with CT rotation measurements (r=0.87). The mean rotation angle at the apex and LIV did not change within five years postoperatively (mean: 0.5±3.6° and 0.4±3.4°, respectively). CSP and DAO were observed in 6.4% and 3.8% of patients, respectively. The inverse trigonometric method is useful to quantitatively evaluate the postoperative rotation angle and identify CSP.

An investigation of Fokas system using two new modifications for the trigonometric and hyperbolic trigonometric function methods

In this work, two new adaptations for the trigonometric and hyperbolic trigonometric function approaches have been presented. These two modifications, entitled modified extended rational sin\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sin$$\\end{document}–cos\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\cos$$\\end{document} function technique and modified extended rational sinh\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sinh$$\\end{document}–cosh\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\cosh$$\\end{document} function method, have been applied for the first time to the Fokas system that represents the nonlinear pulse propagation in monomode fiber optics. We intend to produce innovative, explicit traveling waves, solitons, and periodic wave solutions. These achieved outcomes are presented in the form of exponential functions, trigonometric hyperbolic functions, and combination constructions of the exponential functions along with the trigonometric and hyperbolic trigonometric functions. The obtained solutions reveal significant features of the physical phenomenon and are new. The investigated model incorporates the notions of dispersion, transverse diffusion, degree of dispersion, nonlinear pairing, nonlinear immersion, and the force of the nonlinear interaction among the two components of the system. For the most accurate visual evaluation of the physical importance and dynamic properties, we have presented the findings in a variety of plots, which involve two- and three-dimensional representations. One or more elements in our research that are unique, such as newly modified methodologies, is a new observation that leads researchers to invest in new solutions.

Trigonometric Approximation of the Three-Body Problem: A Double Centroid Approach

_This manuscript introduces the Double Centroid Approximation, a trigonometric method to simplify the Three-body problem in celestial mechanics. By calculating the centroid of three celestial bodies without and then with motion, it offers a novel approach to determine their dynamic center of mass. Grounded in the Superconducting Field Theory, this method simplifies complex gravitational interactions and dark matter considerations. While applicable within our solar system, its principles may extend to other galaxies, highlighting the potential for broad astronomical applications. This concise methodology underscores the importance of triangulation in understanding celestial mechanics, providing a foundation for future research in the field._

Utilizing Inverse Kinematics for Precise Guidance in Planning 6-DoF Robot End-Effector Movements

The solution of the kinematic inverse determines a substantial part of the robotic arm's control accuracy. Researchers frequently employ standard problem-solving techniques such as numerical, algebraic, iterative, and geometric methods. Although geometric like trigonometrical method has been widely studied, and their application is strongly dependent on the shape and dimensions of the robot. The complexity of the steps makes this approach difficult for researchers. In order to give a clearance and easiness, the step-by-step features of inverse kinematics are described in this research. The study begins with forward kinematics and refers to the DH-parameter in Homogeneous Matrix Transformations calculation. The existence of specific elements applied to mathematical derivation constituted the basis of forward kinematic discussions. And based on geometrical analysis, the inverse kinematic is then derived. Furthermore, simulations are performed to demonstrate the actual implementation of IK and the solution is then used to initiate the path planning process.

Optimal modeling of line paths based on multibeam bathymetric system

Multibeam ocean bathymetry is one of the major issues in exploring the ocean. For problems such as multibeam line ocean bathymetry, this paper starts from the related theory, analyzes the inherent geometric features of the problem, comprehensively establishes a multibeam line model with generalized significance within a certain error range, and according to the method of least squares, gives a model of the line with a fixed opening angle of the multibeam transducer in a certain range of the sea area. Firstly, from the geometrical point of view, the interpreted diagram and the schematic diagram given in the title are combined and simplified into a problem of solving a triangle. The trigonometric equations are listed through the sine theorem and solved algebraically. A mathematical model is developed about the coverage width and adjacent strips. Then, from the question, we know that we need to solve the modeling differences caused by the different angles of the measuring lines in different directions, i.e., to elevate the two-dimensional problem of solving triangles to a three-dimensional problem of three-dimensional geometry. By finding the angle between the coverage width and the horizontal plane , we know the relationship between it and the angle between the direction of the survey line and the slope angle , so as to establish the model of the coverage width of multibeam bathymetry. Summarized by the verification of the previous conclusions, this paper obtains: when controlling the width of two adjacent survey lines exactly meets the overlap rate of 10% with the previous survey line, and the survey line direction is perpendicular to the seafloor slope in the projection, the measurement length is the shortest at this time. Using the above summary as a constraint, the shortest path is calculated to be 66 nautical miles using the trigonometric method.

A third-order shear deformation plate bending formulation for thick plates: first principles derivation and applications

A third-order shear deformation plate bending formulation is presented in this study from the first principles. The derivation assumed a displacement field constructed using third-order polynomial function of the transverse (z) coordinate; and made to apriori satisfy the linear three-dimensional (3D) kinematics relations as well as the transverse shear stress free boundary conditions at the top and bottom plate surfaces. The formulation thus has no need for shear stress correction factors of the first-order shear deformation plate theories. The domain equations of equilibrium are obtained as a set of three coupled differential equations in terms of three unknown displacements. The system of coupled equations is solved for simply supported rectangular and square plates subjected to four cases of loading distributions: sinusoidal loading, uniformly distributed loading, linearly distributed loading and point load at the plate center. Navier’s double trigonometric series method is used to construct trial solutions for the three displacement functions such that the boundary conditions are satisfied identically. The integration problem is thus reduced to an algebraic problem and is solved for each considered loading. It is found that the present formulation gives exact results for the normal stresses σxx for sinusoidal and uniformly distributed loads. The study further showed that the results for deflection and stresses agreed with Krishna Murty’s higher order shear deformation plate theory results. The present formulation gave accurate results because of the inclusion of transverse normal strain effects in the formulation. The formulation gives a quadratic variation of the transverse shear stresses across the thickness in consonance with the theory of elasticity method.

Cubic Trigonometric B-spline Method for Solving a Linear System of Second Order Boundary Value Problems

This paper introduces a novel trigonometric B-spline collocation method for solving a specific class of second-order boundary value problems. The study showcases the method’s practicality and effectiveness through various numerical examples. Furthermore, it evaluates the technique’s performance by calculating maximum errors for different step sizes in the spatial domain. The paper also conducts a comparative analysis with alternative methods, demonstrating the superior accuracy of the trigonometric B-spline approach.

Design of optical compensation film for improving the optical performance of organic light emitting diodes

The organic light emitting diode (OLED) display is one of the most promising devices among next-generation information displays because of the advantage of optically clear self-emitting characteristics. Nevertheless, in the conventional OLED display device, the luminous efficiency is increased by using a metal electrode having a high reflectance in the cell, but due to the reflection characteristic in the cell from the incident of external light, the dark characteristic is deteriorated and the contrast characteristic is dramatically reduced. In order to overcome this disadvantage, we propose an optical configuration consisting of a polarizer, quarter-wave plate (QWP), and C-plate for wide-band and wide-view performance, which mean high contrast in visual wavelength range and viewing angle in terms of polar and azimuthal angles while maintaining the transmittance characteristics of OLED when external light is incident. To designing an optimized optical configuration, the phase differences of light were calculated with trigonometric method and polarization characteristics were analyzed using a Poincaré sphere when passing through each optical film. To verify the optical performance of the proposed optical configuration, we applied it to OLED devices through experiments, as a result we confirmed that optical performance could be improved in terms of wide-band and wide-view without disturbing with the transmittance of OLED device.

An explicit analytical description for deep post-buckling behaviors of substrate-free films with complex in-plane biaxial loads

Substrate-free films (i.e., films with in-plane and out-of-plane constraints only on some or all edges) can be found everywhere in nature, daily life, and industrial applications. The theory of their buckling and wrinkling behavior has been mature and widely used in engineering. Significant progress has been made in numerical computation of film instability. However, on the one hand, the inherent drawback of numerical calculations is that they cannot provide explicit results, and the relevant mechanism and parametric analysis are limited; on the other hand, the widely used finite element simulation is difficult to compare the energy of different modes, because the mode is always constrained to a certain order in the simulation (they cannot perform the post-buckling of specified higher order modes as the calculation process is guided by the inherent energy minimization mechanism of the computational frameworks and the higher order modes were excluded). Such two aspects constrain the mechanism study of deep post-buckling related behaviors that dependent on accurate and explicit descriptions of high order modes, e.g., the deep post-buckling bifurcation/secondary buckling. As a result, an explicit analytical description for deep post-buckling behaviors of films is still of significant research value. Reviewing classical explicit analytical descriptions, in any case of the explicit descriptions by multiple trigonometric series method, Galerkin method and perturbation method, in essence, at most three terms of double trigonometric series are used to describe the post-buckling behaviors of thin films explicitly, which always introduces significant error when the film enters deep post-buckling stage. To overcome the stated problems, herein, considering a uniform rectangular film model with arbitrary in-plane loads, and based on the Galerkin method, we firstly develop a high-precision explicit analytical description for the deep post-buckling behaviors of films upon complex in-plane biaxial loads, which proposes a new insight on the buckling mode transition upon extremely deep post-buckling or complex biaxial loads, and suggests that the correct solution of the deep post-buckling bifurcation/secondary buckling requires a complete dynamic model. The research results have theoretical significance on improving the instability mechanical system of thin films, and can also provide theoretical basis for precise control of wrinkle morphology and design of film devices corresponding applications.

Dynamic analysis of 2DFGM porous nanobeams under moving load with surface stress and microstructure effects using Ritz method

This paper investigates the dynamic behavior of micro/nanobeams made of two-dimensional functionally graded porous material (2DFGPM) under accelerated, decelerated, and uniform moving harmonic load, using surface elasticity and modified couple stress theories. The key feature of this formulation is that it deals with a higher order shear deformation beam theory. The non-classical equilibrium equations are developed using Lagrange's equation and the concept of physical neutral surface. The equations of motion are derived using the same approach, accounting for the porosity effect and the modified power-law distribution of material properties. The trigonometric Ritz method is used with sinusoidal trial functions for the displacement field, and the Newmark method is applied to obtain the dynamical response of 2DFGPM nanobeams. The results are compared with previous studies, and the impact of critical parameters such as gradation indices, volume fraction ratio, pattern of porosity, velocity, frequency, and motion type of the applied force are explored. This study highlights the importance of considering the porosity effect, as neglecting it can lead to significant errors in the predicted results. Additionally, the study found that the accelerated and decelerated motions of the applied load have a greater impact on the dynamical deflection of 2DFGPM nanobeams than the uniform motion. The findings of this study can provide guidance for the optimal design of micro/nanobeams subjected to a moving force with multifunctional properties.

Application of the Method of Fast Expansions to Construction of a Trajectory of Movement of a Body with Variable Mass from Its Initial Position in a Gained Final Position in a Gravitational Field

An analytical solution of the problem of the movement of a spacecraft from the starting point to the final point in a certain time is given. First, the method of fast sine expansions is used. The space problem considered here is essentially non-linear, what necessitates the use of trigonometric interpolation methods, which surpass all known interpolations in accuracy and simplicity. In this case, the problem of calculating Fourier coefficients by integral formulas is replaced by the solution of an orthogonal interpolation system. In this regard, two cases are considered on the segment \(\left[ {0,a} \right]\): universal interpolation and trigonometric sine and cosine interpolations. A theorem on the rapid decrease of expansion coefficients is proved, and a compact formula for calculating the interpolation coefficients is obtained. A general theory of fast expansions is given. It is shown that in this case, the Fourier coefficients decrease significantly faster with the growth of the ordinal number compared to the Fourier coefficients in the classical case. This property makes it possible to significantly reduce the number of terms taken into account in the Fourier series, significantly increase the accuracy of calculations and reduce the amount of calculations on a computer. The analysis of the obtained solutions of the spacecraft motion problem is carried out and their comparison with the exact solution of the test problem is proposed. An approximate solution by the method of fast expansions can be taken as an exact one, since the input data of the problem used from reference books have a higher error.

A new statistical methodology using the sine function: Control chart with an application to survival times data.

Statistical methodologies have a wider range of practical applications in every applied sector including education, reliability, management, hydrology, and healthcare sciences. Among the mentioned sectors, the implementation of statistical models in health sectors is very crucial. In the recent era, researchers have shown a deep interest in using the trigonometric function to develop new statistical methodologies. In this article, we propose a new statistical methodology using the trigonometric function, namely, a new trigonometric sine-G family of distribution. A subcase (special member) of the new trigonometric sine-G method called a new trigonometric sine-Weibull distribution is studied. The estimators of the new trigonometric sine-Weibull distribution are derived. A simulation study of the new trigonometric sine-Weibull distribution is also provided. The applicability of the new trigonometric sine-Weibull distribution is shown by considering a data set taken from the biomedical sector. Furthermore, we introduce an attribute control chart for the lifetime of an entity that follows the new trigonometric sine-Weibull distribution in terms of the number of failure items before a fixed time period is investigated. The performance of the suggested chart is investigated using the average run length. A comparative study and real example are given for the proposed control chart. Based on our study of the existing literature, we did not find any published work on the development of a control chart using new probability distributions that are developed based on the trigonometric function. This surprising gap is a key and interesting motivation of this research.