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Related Concept Videos

Limits on Trigonometric Functions01:25

Limits on Trigonometric Functions

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Limits on Trigonometric FunctionsThe limits of trigonometric functions play a fundamental role in calculus, particularly in defining derivatives. One of the most important results is:which is important for differentiating trigonometric functions and is widely applied in mathematical analysis and physics.Geometric IntuitionA common approach to proving this result involves analyzing a sector of a unit circle with an angle subtended at the center. Since the arc length is numerically equal to the...
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The motion of a Ferris wheel rotating at a constant speed provides an intuitive model for understanding trigonometric functions and their derivatives. As a rider moves along the circular path, the vertical height above the ground changes smoothly and periodically over time. This vertical motion can be accurately represented by a sine function, reflecting the repeating pattern of ascent and descent inherent to circular motion.Height and Rate of ChangeIf the rider’s height is modeled by a...
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Trigonometric functions exhibit periodic and symmetrical behavior, deeply rooted in the unit circle. The sine and cosine functions correspond to the vertical and horizontal projections, respectively, of a point rotating counterclockwise around the circle. These functions trace smooth, repeating waveforms with identical periods and bounded ranges. The tangent function is defined as the ratio of sine to cosine and produces an unbounded curve that repeats every units, with vertical asymptotes...
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When observing the vertical ascent of an object from a fixed ground position, such as a rocket launch, trigonometric relationships offer a precise method for determining the object's height. As the object rises, an observer stationed at a known horizontal distance from the launch site can measure the angle between the ground and the object's current position. This dynamic angle provides critical information that connects the observed position with its height above the ground.The tangent...
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Inverse trigonometric functions are fundamental mathematical tools that reverse the actions of standard trigonometric functions. While trigonometric functions map angles to ratios, inverse trigonometric functions perform the opposite operation by mapping a ratio back to its corresponding angle. These functions are essential in various applications, particularly in determining angles when given specific distances, such as calculating elevation angles in navigation and engineering.For a function...
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The unit circle—a circle with a radius of one, centered at the origin of the coordinate plane—serves as the foundational framework for defining trigonometric functions. In this context, arc length refers to the distance measured along the circumference of the circle between two points, and it provides a way to represent real numbers geometrically. Each real number t corresponds to an arc length measured counterclockwise from the positive x-axis around the circle. The coordinates of...
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A two-point-Padé-approximant-based method for bounding some trigonometric functions.

Xiao-Diao Chen1, Junyi Ma1, Jiapei Jin1

  • 11Key Laboratory of Complex Systems Modeling and Simulation, Hangzhou Dianzi University, Hangzhou, China.

Journal of Inequalities and Applications
|August 24, 2018
PubMed
Summary
This summary is machine-generated.

This study introduces a novel two-point Padé approximant method for finding and proving bounds in engineering inequalities, specifically Wilker type inequalities. The approach improves upon existing methods and offers potential applications for a broader range of inequalities.

Keywords:
Becker–Stark’s inequalityPadé approximantTrigonometric approximationTwo-sided boundsWilker’s inequality

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Area of Science:

  • Engineering Mathematics
  • Numerical Analysis

Background:

  • Bounding inequalities are crucial for solving engineering problems.
  • Key challenges include determining bounds and rigorously proving them.

Purpose of the Study:

  • To present a novel method for finding and proving bounds for Wilker type inequalities.
  • To demonstrate the method's effectiveness and potential for broader applications.

Main Methods:

  • Utilizing a two-point Padé approximant-based approach.
  • Developing a new technique for proving the established bounds.

Main Results:

  • Successfully recovered existing estimates from Mortici's method.
  • Achieved new and improved estimates compared to prevailing methods.
  • Demonstrated the method's applicability to Wilker type inequalities.

Conclusions:

  • The proposed two-point Padé approximant method offers an effective way to address bounding inequality challenges.
  • This technique provides enhanced estimates and can be extended to other types of inequalities in engineering and mathematics.