Unit Circles and Inverse Trigonometric Functions

2014 ◽  
Vol 108 (2) ◽  
pp. 114-119
Author(s):  
Azael Barrera
Keyword(s):  
Unit Circle ◽  
Trigonometric Functions ◽  
Unit Circles ◽  
Inverse Trigonometric Functions

A method to determine all the inverse trigonometric functions directly from the unit circle.

scholarly journals The chord-length distribution of a polyhedron

2020 ◽  
Vol 76 (4) ◽  
pp. 474-488
Author(s):  
Salvino Ciccariello
Keyword(s):  
Distribution Function ◽  
Length Distribution ◽  
Chord Length ◽  
Square Root ◽  
Trigonometric Functions ◽  
Chord Length Distribution ◽  
Analytic Expressions ◽  
Analytic Expression ◽  
Positive Integers ◽  
Inverse Trigonometric Functions

The chord-length distribution function [γ′′(r)] of any bounded polyhedron has a closed analytic expression which changes in the different subdomains of the r range. In each of these, the γ′′(r) expression only involves, as transcendental contributions, inverse trigonometric functions of argument equal to R[r, Δ1], Δ1 being the square root of a second-degree r polynomial and R[x, y] a rational function. As r approaches δ, one of the two end points of an r subdomain, the derivative of γ′′(r) can only show singularities of the forms |r − δ|−n and |r − δ|−m+1/2, with n and m appropriate positive integers. Finally, the explicit analytic expressions of the primitives are also reported.

scholarly journals Gaussian Qualitative Trigonometric Functions in a Fuzzy Circle

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
M. Clement Joe Anand ◽  
Janani Bharatraj
Keyword(s):  
Particle Swarm Optimization ◽  
Unit Circle ◽  
Fuzzy Theory ◽  
Particle Swarm ◽  
Membership Functions ◽  
Trigonometric Functions ◽  
Swarm Optimization ◽  
Qualitative Representation ◽  
Trigonometric Identities

We build a bridge between qualitative representation and quantitative representation using fuzzy qualitative trigonometry. A unit circle obtained from fuzzy qualitative representation replaces the quantitative unit circle. Namely, we have developed the concept of a qualitative unit circle from the view of fuzzy theory using Gaussian membership functions, which play a key role in shaping the fuzzy circle and help in obtaining sharper boundaries. We have also developed the trigonometric identities based on qualitative representation by defining trigonometric functions qualitatively and applied the concept to fuzzy particle swarm optimization using α-cuts.

The Ambiguity of the Inverse Secant

2018 ◽  
Vol 111 (6) ◽  
pp. 470-475
Author(s):  
Thomas J. Bannon
Keyword(s):  
Trigonometric Functions ◽  
Definition Of ◽  
Inverse Trigonometric Functions

Defining inverse trigonometric functions involves choosing ranges for the functions. The choices made for the inverse sine, cosine, tangent, and cotangent functions follow generally accepted conventions. However, different authors make different choices when defining y = arcsec x and y = arccsc x for negative x. I first discovered that the definitions of these functions were not a settled convention when I found an alternate definition in Schaum's (Ayers and Mendelson 2012) and Anton's (1995) books. The more commonly used definition is simpler and results in a function more easily evaluated and for that reason is preferable when introducing the inverse trigonometric functions in an algebra or precalculus course. As we shall see, though, the alternate definition of the inverse secant function has many advantages when we move on to calculus. Since we have a choice in our definitions, we should choose what makes the most sense in context.

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