Inverse trigonometric functions

About: Inverse trigonometric functions is a research topic. Over the lifetime, 854 publications have been published within this topic receiving 11141 citations. The topic is also known as: arcus function & antitrigonometric function.

Golden Ratio Base Expansions of the Logarithm and Inverse Tangent of Fibonacci and Lucas Numbers

Abstract: Let $\alpha=(1+\sqrt 5)/2$, the golden ratio, and $\beta=-1/\alpha=(1 - \sqrt 5)/2$. Let $F_n$ and $L_n$ be the Fibonacci and Lucas numbers, defined by $F_n=(\alpha^n -\beta^n)/\sqrt 5$ and $L_n=\alpha^n + \beta^n$, for all non-negative integers. We derive base~$\alpha$ expansions of $\log F_n$, $\log L_n$, $\arctan\dfrac1{F_n}$ and $\arctan\dfrac1{L_n}$ for all positive integers $n$.

Some New Expressions of Compactons in Compressible Elastic Rods

Abstract: In this paper, we use phase plane analysis to study the compactons of the nonlinear equation. Four new implicit expressions of the compactons are obtained. These new implicit expressions are given by inverse tangent functions. Our work extends previous results. For two sets of the data, the graphs of the implicit functions are drawn and numerical simulations are given to test the correctness of our theoretical results.

Probabilistic Interpolation of the Curve via the Method of Hurwitz-Radon Matrices

Abstract: Mathematics and computer science are interested in methods of curve interpolation using the set of key points (knots). A proposed method of Hurwitz- Radon Matrices (MHR) is such a method. This novel method is based on the family of Hurwitz-Radon (HR) matrices which possess columns composed of orthogonal vectors. Two-dimensional curve is interpolated via different functions as probability distribution functions: polynomial, sinus, cosine, tangent, cotangent, logarithm, exponent, arcsin, arccos, arctan, arcctg or power function, also inverse functions. It is shown how to build the orthogonal matrix operator and how to use it in a process of curve reconstruction.

How to Explain The Empirical Success of Generalized Trigonometric Functions in Processing Discontinuous Signals

Abstract: Fourier series, that represent a signal as a linear combination of sinusoids, are efficiently used in science and engineering to approximate smooth signals. However, for discontinuous signals that describe abrupt transitions – such as phase transitions, earthquakes, etc., Fourier approximations lead to Gibbs phenomenon: large oscillations near the discontinuity. It is possible to avoid these oscillations if, instead of sinusoids, we use discontinuous functions – e..g., Haar wavelets – but the resulting representation is not very computationally efficient for smooth signals. It is therefore desirable to come up with a representation which would be efficient both for smooth and for discontinuous signals. A successful semi-heuristic approach to solving this problem is the use of generalized trigonometric functions instead of the sinusoids. Specifically, sin(x)

Time measurement circuit

Abstract: PROBLEM TO BE SOLVED: To measure time over a wide dynamic range, with high stability and a high accuracy. SOLUTION: A standard clock synchronous to a trigonometric function wave of a sine wave and a cosine wave is generated at a standard pulse generation circuit 12. Two kinds of the trigonometric function waves are A/D converted by ADC 16, 18 with a timing of arrival of a start signal, and this value is subjected to inverse trigonometric function operation by an operation processing device 46 to calculate a micro time at a start side. A sine time at a stop side is also calculated similarly. A pulse is generated at an AND circuit 40 between synchronous start and stop signals taking a synchronism at FF 24, 38 and is counted by a scaler 42 and a count number at the time of stop is stored in a register. The operation processing device 46 calculates the overall measurement time using the count number, clock period time and two micro times.