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Kinematics in Linear Motion
Published in Emeric Arus, Biomechanics of Human Motion, 2017
A trigonometric equation is an equation consisting of trigonometric functions. Trigonometric identity expressing a trigonometric equation that becomes true when the variable is replaced by every permissible number. Trigonometric ratio is a ratio that describes a relationship between a side and an angle of a triangle. Inverse trigonometric functions are the inverse functions of the trigonometric functions, written sin−1 (arcsin), cos−1 (arccos), tan−1 (arctan), cot−1 (arccot), sec−1 (arcsec), and csc−1 (arccsc). Here, the author will describe the trigonometric functions using the triangle shown by Figure 6.9.
Chapter S1: Elementary Functions and Their Properties
Published in Andrei D. Polyanin, Valentin F. Zaitsev, Handbook of Ordinary Differential Equations, 2017
Andrei D. Polyanin, Valentin F. Zaitsev
Inverse trigonometric functions (arc functions) are the functions that are inverse to the trigonometric functions. Since the trigonometric functions sinx, $ {\text{sin }}x, $ cosx, $ {\text{cos }}x, $ tanx, $ {\text{tan }}x, $ cotx $ {\text{cot }}x $ are periodic, the corresponding inverse functions, denoted by Arcsinx, $ Arc {\text{sin }}x, $ Arccosx, $ Arc {\text{cos }}x, $ Arctanx, $ Arc {\text{tan }}x, $ Arccot x, are multi-valued. The following relations define the multi-valued inverse trigonometric functions:
A self-enforced optimal framework for inter-platoon transfer in connected vehicles
Published in Journal of Intelligent Transportation Systems, 2023
In platooning, the major preference is always given to the destination by any PM. It is also of great importance that any platoon of interest should be prone to minimal disruption in transit. Disruption can be caused by the presence of any PM having a different destination than the respective PL in a platoon which might migrate. The preference degree of the Platoon to the PL based on the destination can be calculated as:where is the number of PMs in platoon p designated to the destination same as that of the respective PL. According to the nature of inverse trigonometric functions, with the increase of the independent variable increases monotonously and approaches 1 infinitely, which conforms to the basis that the preference degree increases as the number of members with a similar destination to that of the PL increases, owing to lesser disruption.
Rotation sequence to report humerothoracic kinematics during 3D motion involving large horizontal component: application to the tennis forehand drive
Published in Sports Biomechanics, 2018
Thomas Creveaux, Violaine Sevrez, Raphaël Dumas, Laurence Chèze, Isabelle Rogowski
Twelve Euler/Cardan rotation sequences exist, based on the ordering of three rotations about an axis of the proximal segment, a floating axis and an axis of the distal segment. Care must therefore be taken when choosing a particular rotation sequence, as matrix multiplication is not commutative and several studies have documented significant differences in angular kinematics between various angle conventions and sequences (Bonnefoy-Mazure et al., 2010; Karduna et al., 2000; Phadke et al., 2011; Senk & Chèze, 2006; Sinclair, Taylor, Edmundson, Brooks, & Hobbs, 2012). Indeed, each of the three-rotation permutations differently defines the magnitudes of the three rotation angles, possibly leading to incoherent joint amplitude. The decomposition occurring in Euler/Cardan angles may further lead to two other types of singularities in the time histories of the angle values (van der Helm, 1997). First, when approaching the so-called ‘gimbal lock’ position (i.e. when the first and third rotation axes coincide), the computed angle values become very sensitive to joint motion, resulting in physiologically meaningless angle variations. Second, the necessary use of inverse trigonometric functions can result in phase angle discontinuities, characterised by large absolute differences – up to 360° – between two successive angle values.
The impact of procedural and conceptual teaching on students' mathematical performance over time
Published in International Journal of Mathematical Education in Science and Technology, 2021
Vahid Borji, Farzad Radmehr, Vicenç Font
In relation to D5, fifteen students of each group showed D5 in their responses to the Q2.c of Test 1. They considered equals to but could not find using differentiation correctly. This problem was related to a lack of knowledge about derivatives of inverse trigonometric functions (Figure 4.B). For instance, in Figure 4.B, the student computed incorrectly the derivative of equals to .