Some Applicable Methods Of Approximating Basic Trigonometric Functions And Their Inverse Value

Abstract: This paper presents some applicable methods of approximating basic trigonometric functions and their inverse value. Methods are the best choice when a need arise to know first few digits after a decimal point and corresponding angle without spending time for immediate purpose. The ways of approximation are helpful for science and engineering field of study; they can be applied to get immediate solutions for practical problems which might be estimating, comparing and judging while operations of numbers. The assumption stated to carry out this work is; There exists certain function which can satisfies the condition defined as; if the sequence of some domain values within a domain of function forms an arithmetic progression, then the sequence of corresponding range values within a range of function will also forms an arithmetic progression. This assumption leads to the assumed generalized approximate equation and finally to the major findings. The major areas of study to carry out this particular work are arithmetic progression, sine function, cosine function and idea related to trigonometric functions such as trigonometric identities, co terminal angles, reference angle and co function definition. The objective is to contribute additional alternative knowledge to the Mathematical science. The findings of this paper are useful to derive general approximation formulae and other related findings that will be presented in the future

Improving the Modified Universal Soil Loss Equation by Physical Interpretation of Its Factors

A primary objective of this paper is to change the input data requirement of the Modified Universal Soil Loss Equation (MUSLE) for the calculation of its runoff factor for possible application in data-scarce areas. Basically, the MUSLE was developed for a small agricultural watershed, where the extent of erosion is from sheet to rill erosion, but we cannot exactly tell whether it considers gully erosion or not. The underlying physical assumption to improve the MUSLE is that the amount of potential energy of runoff is proportional to the shear stress for sediment transport from a slope field and the kinetic energy of the runoff at the bottom of the slope field for gully formation. The improved MUSLE was tested at four watersheds in Ethiopia, and it showed better performance (i.e., the minimum performance is 84%) over the original MUSLE (i.e., the minimum performance was 80%), for all four watersheds under our consideration. We expect the same to be true for other watersheds of Ethiopia

An iterative approach for deriving and solving an accurate regression equation

ABSTRACTThis paper introduces a method for deriving an accurate regression equation based on a set of any paired data, and a technique for solving the equation. For a practical example, we used five hundred seventy-one pairs of sediment concentration and river flow data to derive an accurate sediment rating equation. The graphs of the measured and predicted sediment concentrations matched each other, and data correlation showed Nash–Sutcliffe efficiency (NSE) of 0.9999860, coefficient of determination ([Formula: see text]) of 0.99998679, root mean square error (RMSE) of 0.0345, mean average error (MAE) of 0.0067, volume error (VE) of 1, and sum of square error (SSE) of 0.678631. To explain the technique of deriving and solving the accurate regression equation, separate files of video presentation and excel spreadsheet are provided as supplementary materials. In general, the method can be used to model any processes, and any calibration and validation processes can be addressed

Estimating the Best Exponent and the Best Combination of the Exponent and Topographic Factor of the Modified Universal Soil Loss Equation under the Hydro-Climatic Conditions of Ethiopia

The effect of the topographic factor of the Modified Universal Soil Equation (MUSLE) on soil erosion and sediment yield is not clear. Except for the coefficient, soil erodibility, cover, and conservation practice factors of the MUSLE, an individual effect of the exponents and topographic factors of the MUSLE on soil erosion and sediment yield can be seen by applying the model at different watersheds. A primary objective of this paper is to estimate the best exponents and topographic factors of the MUSLE under the hydro-climatic conditions of Ethiopia. For the sake of the calibration procedure, the main factors of the MUSLE that directly affect the soil erosion process, such as cover, conservation practice, soil erodibility, and topographic factors, are estimated based on past experiences from the literature and comparative approaches, whereas the parameters that do not directly affect the erosion process or that have no direct physical meaning (i.e., coefficient a and exponent b) are estimated through calibration. We verified that the best exponent of the MUSLE is 1 irrespective of the topographic factor, which results in the maximum performance of the MUSLE (i.e., approximately 100%). The best exponent that corresponds to the best equation of the topographic factor is 0.57; in this case, the performance of the model is greater than or equal to 80% for all watersheds under our consideration. We expect the same for other watersheds of Ethiopia, while for other exponents and topographic factors, the performance of the model decreases. Therefore, for the conditions of Ethiopia, the original exponent of the MUSLE is changed from 0.56 to 0.57, and the best equations of the topographic factor are provided in this paper

Improving the Modified Universal Soil Loss Equation by Physical Interpretation of Its Factors

A primary objective of this paper is to change the input data requirement of the Modified Universal Soil Loss Equation (MUSLE) for the calculation of its runoff factor for possible application in data-scarce areas. Basically, the MUSLE was developed for a small agricultural watershed, where the extent of erosion is from sheet to rill erosion, but we cannot exactly tell whether it considers gully erosion or not. The underlying physical assumption to improve the MUSLE is that the amount of potential energy of runoff is proportional to the shear stress for sediment transport from a slope field and the kinetic energy of the runoff at the bottom of the slope field for gully formation. The improved MUSLE was tested at four watersheds in Ethiopia, and it showed better performance (i.e., the minimum performance is 84%) over the original MUSLE (i.e., the minimum performance was 80%), for all four watersheds under our consideration. We expect the same to be true for other watersheds of Ethiopia

An iterative approach for deriving and solving an accurate regression equation

This paper introduces a method for deriving an accurate regression equation based on a set of any paired data, and a technique for solving the equation. For a practical example, we used five hundred seventy-one pairs of sediment concentration and river flow data to derive an accurate sediment rating equation. The graphs of the measured and predicted sediment concentrations matched each other, and data correlation showed Nash–Sutcliffe efficiency (NSE) of 0.9999860, coefficient of determination (R2) of 0.99998679, root mean square error (RMSE) of 0.0345, mean average error (MAE) of 0.0067, volume error (VE) of 1, and sum of square error (SSE) of 0.678631. To explain the technique of deriving and solving the accurate regression equation, separate files of video presentation and excel spreadsheet are provided as supplementary materials. In general, the method can be used to model any processes, and any calibration and validation processes can be addressed.</p