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In this paper, we establish two new inequalities of the Masjed Jamei type for inverse trigonometric and inverse hyperbolic functions and apply them to obtain some refinement and extension of Mitrinović–Adamović and Lazarević inequalities. The inequalities obtained in this paper go beyond the conclusions and conjectures in the previous literature. Finally, we apply the main results of this paper to the field of mean value inequality and obtain two new inequalities on Seiffert-like means and classical means.

We establish new simple bounds for the quotients of inverse trigonometric and inverse hyperbolic functions such as sin−1xsinh−1x and tanh−1xtan−1x. The main results provide polynomial bounds using even quadratic functions and exponential bounds under the form eax2. Graph validation is also performed.

In this paper, we present Shafer-type inequalities for inverse trigonometric functions and Gauss lemniscate functions.

Spherical trigonometry was at the heart of astronomy and ocean-going navigation for two millennia. The discipline was a mainstay of mathematics education for centuries, and it was a standard subject in high schools until the 1950s. Today, however, it is rarely taught.Heavenly Mathematicstraces the rich history of this forgotten art, revealing how the cultures of classical Greece, medieval Islam, and the modern West used spherical trigonometry to chart the heavens and the Earth. Glen Van Brummelen explores this exquisite branch of mathematics and its role in ancient astronomy, geography, and cartography; Islamic religious rituals; celestial navigation; polyhedra; stereographic projection; and more. He conveys the sheer beauty of spherical trigonometry, providing readers with a new appreciation for its elegant proofs and often surprising conclusions. Heavenly Mathematicsis illustrated throughout with stunning historical images and informative drawings and diagrams that have been used to teach the subject in the past. This unique compendium also features easy-to-use appendixes as well as exercises at the end of each chapter that originally appeared in textbooks from the eighteenth to the early twentieth centuries.

Due to the great success of hypergeometric functions of one variable, a number of hypergeometric functions of two or more variables have been introduced and explored. Among them, the Kampé de Fériet function and its generalizations have been actively researched and applied. The aim of this paper is to provide certain reduction, transformation and summation formulae for the general Kampé de Fériet function and Srivastava’s general triple hypergeometric series, where the parameters and the variables are suitably specified. The identities presented in the theorems and additional comparable outcomes are hoped to be supplied by the use of computer-aid programs, for example, Mathematica. Symmetry occurs naturally in p+1Fp, the Kampé de Fériet function and the Srivastava’s function F(3)[x,y,z], which are three of the most important functions discussed in this study.

This chapter contains sections titled: Introduction Trigonometric Functions (With Restricted Domains) and Their Inverses The Inverse Cosine Function The Inverse Tangent Function Definition of the Inverse Cotangent Function Formula for the Derivative of Inverse Secant Function Formula for the Derivative of Inverse Cosecant Function Important Sets of Results and their Applications Application of Trigonometric Identities in Simplification of Functions and Evaluation of Derivatives of Functions Involving Inverse Trigonometric Functions

An unusual alternating reflection method on conics is presented to evaluate inverse trigonometric and hyperbolic functions.

The Padé approximation is a useful method for creating new inequalities and improving certain inequalities. In this paper we use the Padé approximant to give the refinements of some remarkable inequalities involving inverse trigonometric functions, it is shown that the new inequalities presented in this paper are more refined than that obtained in earlier papers.

Purpose The crankshaft phenomenon (CSP) is a corrective loss after posterior surgery for early onset scoliosis (EOS). However, an accurate method for CSP evaluation has yet to be developed. In this study, we evaluated pedicle screw (PS) length and rotation angle using an inverse trigonometric function and investigated the prevalence of the CSP. Methods Fifty patients from nine institutions (mean age 10.6 years, male/female ratio 4:46) who underwent early definitive fusion surgery at ≤ 11 years of age were included. The rotation angle was calculated as arctan (lateral/frontal PS length) using radiography. Measurements were taken at the apex and lower instrumented vertebra (LIV) immediate, 2-, and 5-year postoperatively. CSP was defined as a rotation angle progression ≥ 5°. We divided patients into CSP and non-CSP groups and measured the demographic parameters, Risser grade, state of the triradiate cartilage, major coronal Cobb angle, T1–T12 length, T1–S1 length, and presence of distal adding-on (DAO). We compared these variables between groups and investigated the correlation between the measured variables and vertebral rotation. Logistic regression analysis investigated factors associated with CSP. Results The rotation angle progressed by 2.4 and 1.3° over 5 years for the apex and LIV, respectively. CSP occurred in 15 cases (30%), DAO in 11 cases (22%), and CSP and DAO overlapped in 4 cases (8%). In the CSP group, the T1–T12 length was low immediate postoperatively. The rotation angle was negatively correlated with preoperative height (r = − 0.33), T1–T12 length (r = − 0.35), and T1–S1 length (r = − 0.30). A lower preoperative T1–T12 length was associated with CSP (odds ratio: 0.996, p = 0.048). Conclusions CSP occurred in 30% of patients with EOS who underwent definitive fusion. The presence of CSP was associated with a lower preoperative T1–T12 length. Level of evidence Diagnosis, level IV.