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Engineering and Scientific Calculations
Published in David E. Clough, Steven C. Chapra, Introduction to Engineering and Scientific Computing with Python, 2023
David E. Clough, Steven C. Chapra
The trigonometric functions relate the angles in a right triangle to its sides and hypotenuse. They are often conveniently represented on a two-dimensional graph using a unit circle, that is, a circle with a radius of 1. Compare the definitions in Equation 1.11 to the diagram in Figure 1.2.sin(θ)=bcos(θ)=atan(θ)=ba
Math Tools
Published in Thomas M. Nordlund, Peter M. Hoffmann, Quantitative Understanding of Biosystems, 2019
Thomas M. Nordlund, Peter M. Hoffmann
Trigonometric functions commonly occur as mathematical solutions to equations found in physics, chemistry, biology, microscopy (imaging), and other fields. If the angles in these functions are small, about 10° (~0.17 rad) or less, the common trig functions can be approximated:
Viscoelastic damped response of laminated composite shells subjected to various dynamic loads
Published in Mechanics Based Design of Structures and Machines, 2023
Orders greater than (z/R)2 are truncated (Leissa and Chang 1996). In this way, Eq. (20) is simplified to the following equations: where the parameter Ki and Kj are defined as the shear correction factors. K is used as 5/6 in this case (Timoshenko 1921). Furthermore, in the summing term, “k” denotes the counter for layer. The Co value and mass moment inertia terms are as follows: where (k) denotes the mass density of the shell's kth layer per unit midsurface region. The Navier solution is applicable to both thin and thick shells. The displacement part of the shells are assumed to be represented by trigonometric functions of sine and cosine in this solution type. The shell's edges are assumed to have shear diaphragm boundaries. The boundary conditions for simply supported thick shells are as follows:
A Novel Quasi-Oppositional Learning-Based Chaos-Assisted Sine Cosine Algorithm for Hybrid Energy Integrated Dynamic Economic Emission Dispatch
Published in IETE Journal of Research, 2023
Koustav Dasgupta, Provas Kumar Roy, Vivekananda Mukherjee
A novel SCA algorithm, proposed by Mirjalili [42], has been discussed in this section. The SCA algorithm is mainly worked by sine and cosine function. Mirjalili introduced trigonometric functions, i.e., sine and cosine functions, to optimize the problem of real-world optimization problems. This algorithm involves the sine and the cosine functions. It is used to update initial set of the population to find better results on the basis of fitness value. In this section, a new modified SCA has been introduced by improving performance of basic SCA. In the current article, this population-based SCA algorithm has been modified using QOL strategy surfaced in Ref. [49] for improving the performance of the basic SCA. The concept of chaos [47] has been introduced in SCA technique for balancing between the exploration and the exploitation phases. This hybridization helps to improve the searching ability of the SCA and helps to bring better results. The primary idea behind QOL is to evaluate an estimate and its inverse estimate at the same time in order to improve the accuracy of the present feasible solution. Moreover, the scheme of jumping factor (JF) has been used to find higher convergence rate and high-quality solution by jumping between initial and opposite population set. Good local searching ability of the algorithm is very essential for obtaining the desired goal. Here, QOL-CSCA consists of four steps, i.e., initialization, mutation, recombination and selection. The optimization process of the SCA is explained below step by step.
Integrating rational functions of sine and cosine using the rules of Bioche
Published in International Journal of Mathematical Education in Science and Technology, 2022
To help ease this burden a number of standard techniques are available allowing one to integrate a large class of functions. One such class is the class of rational functions. Students discover how every rational function has a primitive among the elementary functions. And what is truly amazing, there is a systematic way of finding the primitive using the method of partial fractions. By extension, integrals consisting of rational functions of the six trigonometric functions can always be found by the use of a rationalizing substitution which converts a rational function of sine and cosine (only these two trigonometric functions need to be considered since the other four trigonometric functions can all be expressed in terms of sine and cosine) into a rational function in terms of the substitution variable used. Such a method, while often found in older calculus texts (Edwards, 1921, p. 188), today tends to be relegated to one of the exercises found in the end of chapter exercise sets (Stewart, 2008, p. 517), (Spivak, 2008, pp. 386–387), if given at all.