The document discusses inverse trigonometric functions. It defines the inverse functions of sine, cosine, tangent, cotangent, secant and cosecant by restricting their domains such that they become one-to-one functions. The principal value branches are defined for each inverse function, with their domains and ranges specified. Graphs are provided to illustrate the inverse relationships between the trigonometric functions and their inverses. Properties like how to obtain the graph of an inverse function from the original are also outlined.
1. The document discusses functions and their notation. It defines functions as mappings from inputs to outputs and provides examples of function notation.
2. It also discusses different types of relations between inputs and outputs such as one-to-one, one-to-many, many-to-one, and many-to-many relations.
3. The document explains concepts related to functions such as composite functions, inverse functions, and how to determine composite and inverse functions through examples.
This document provides an overview of functions from chapter 1 of an additional mathematics module. It defines key terms like domain, codomain, range, and discusses different types of relations including one-to-one, many-to-one, and many-to-many. It also covers function notation, evaluating functions, composite functions, and provides examples of calculating images and objects of functions. The chapter aims to introduce students to the fundamental concepts of functions through definitions, diagrams, and practice exercises.
Abstract: The Riemann zeta function is one of the most Leonhard Euler important and fascinating functions in mathematics. Analyzing the matter of conjecture of Riemann divide our analysis in the zeta function and in the proof of conjecture, which has consequences on the distribution of prime numbers.
This document discusses three theorems about Sylow subgroups in finite groups. Theorem 1 proves the existence of a group with q^e Sylow p-subgroups, where q and p are primes and q^e ≡ 1 (mod p). Theorem 2 shows that if p and q are "mod-1 related", meaning q ≡ 1 (mod p), then there exists a group with q^n Sylow p-subgroups for any n. Theorem 3 deals specifically with the 2-case, proving there exists a group with n Sylow 2-subgroups for any positive odd integer n. The document provides constructions of groups to satisfy the conditions of each theorem and proofs of subsidiary lemmas about the properties of
This document discusses three theorems about Sylow subgroups in finite groups. Theorem 1 proves the existence of a group with q^e Sylow p-subgroups, where q and p are primes and q^e ≡ 1 (mod p). Theorem 2 shows that if p and q are "mod-1 related", meaning q ≡ 1 (mod p), then there exists a group with q^n Sylow p-subgroups for any n. Theorem 3 deals specifically with the 2-case, proving there exists a group with n Sylow 2-subgroups for any positive odd integer n. The document provides constructions of groups to satisfy the conditions of each theorem and proofs of subsidiary lemmas about the properties of
Best Approximation in Real Linear 2-Normed SpacesIOSR Journals
This pape r d e l i n e a t e s existence, characterizations and st rong unicity of best uniform
approximations in real linear 2-normed spaces.
AMS Su ject Classification: 41A50, 41A52, 41A99, 41A28.
The document provides a review outline for Math 1a Midterm II covering topics including: differentiation using product, quotient, and chain rules; implicit differentiation; logarithmic differentiation; applications such as related rates and optimization; and the shape of curves including the mean value theorem and extreme value theorem. It also lists learning objectives and provides details on key concepts like L'Hopital's rule and the closed interval method for finding extrema.
Plan: 1. Classification of general polyadic systems and special elements. 2. Definition of n-ary semigroups and groups. 3. Homomorphisms of polyadic systems. 4. The Hosszú-Gluskin theorem and its “q-deformed” generalization. 5. Multiplace generalization of homorphisms - heteromorpisms. 6. Associativity quivers. 7. Multiplace representations and multiactions. 8. Examples of matrix multiplace representations for ternary groups. 9. Polyadic rings and fields. 10. Polyadic analogs of the integer number ring Z and the Galois field GF(p). 11. Equal sums of like powers Diophantine equation over polyadic integer numbers.
1. The document discusses functions and their notation. It defines functions as mappings from inputs to outputs and provides examples of function notation.
2. It also discusses different types of relations between inputs and outputs such as one-to-one, one-to-many, many-to-one, and many-to-many relations.
3. The document explains concepts related to functions such as composite functions, inverse functions, and how to determine composite and inverse functions through examples.
This document provides an overview of functions from chapter 1 of an additional mathematics module. It defines key terms like domain, codomain, range, and discusses different types of relations including one-to-one, many-to-one, and many-to-many. It also covers function notation, evaluating functions, composite functions, and provides examples of calculating images and objects of functions. The chapter aims to introduce students to the fundamental concepts of functions through definitions, diagrams, and practice exercises.
Abstract: The Riemann zeta function is one of the most Leonhard Euler important and fascinating functions in mathematics. Analyzing the matter of conjecture of Riemann divide our analysis in the zeta function and in the proof of conjecture, which has consequences on the distribution of prime numbers.
This document discusses three theorems about Sylow subgroups in finite groups. Theorem 1 proves the existence of a group with q^e Sylow p-subgroups, where q and p are primes and q^e ≡ 1 (mod p). Theorem 2 shows that if p and q are "mod-1 related", meaning q ≡ 1 (mod p), then there exists a group with q^n Sylow p-subgroups for any n. Theorem 3 deals specifically with the 2-case, proving there exists a group with n Sylow 2-subgroups for any positive odd integer n. The document provides constructions of groups to satisfy the conditions of each theorem and proofs of subsidiary lemmas about the properties of
This document discusses three theorems about Sylow subgroups in finite groups. Theorem 1 proves the existence of a group with q^e Sylow p-subgroups, where q and p are primes and q^e ≡ 1 (mod p). Theorem 2 shows that if p and q are "mod-1 related", meaning q ≡ 1 (mod p), then there exists a group with q^n Sylow p-subgroups for any n. Theorem 3 deals specifically with the 2-case, proving there exists a group with n Sylow 2-subgroups for any positive odd integer n. The document provides constructions of groups to satisfy the conditions of each theorem and proofs of subsidiary lemmas about the properties of
Best Approximation in Real Linear 2-Normed SpacesIOSR Journals
This pape r d e l i n e a t e s existence, characterizations and st rong unicity of best uniform
approximations in real linear 2-normed spaces.
AMS Su ject Classification: 41A50, 41A52, 41A99, 41A28.
The document provides a review outline for Math 1a Midterm II covering topics including: differentiation using product, quotient, and chain rules; implicit differentiation; logarithmic differentiation; applications such as related rates and optimization; and the shape of curves including the mean value theorem and extreme value theorem. It also lists learning objectives and provides details on key concepts like L'Hopital's rule and the closed interval method for finding extrema.
Plan: 1. Classification of general polyadic systems and special elements. 2. Definition of n-ary semigroups and groups. 3. Homomorphisms of polyadic systems. 4. The Hosszú-Gluskin theorem and its “q-deformed” generalization. 5. Multiplace generalization of homorphisms - heteromorpisms. 6. Associativity quivers. 7. Multiplace representations and multiactions. 8. Examples of matrix multiplace representations for ternary groups. 9. Polyadic rings and fields. 10. Polyadic analogs of the integer number ring Z and the Galois field GF(p). 11. Equal sums of like powers Diophantine equation over polyadic integer numbers.
It turns out that an equation x^p+y^q=z^w has a solution in Z+ if and only if at least one exponent p or q or w equals 2.DOI:10.13140/RG.2.2.15459.22567/7
Fixed point result in menger space with ea propertyAlexander Decker
This document presents a fixed point theorem for four self-maps in a Menger space. It begins by defining key concepts related to Menger spaces including probabilistic metric spaces, t-norms, neighborhoods, convergence, Cauchy sequences, and completeness. It then introduces properties like weakly compatible maps, property (EA), and JSR mappings. The main result, Theorem 3.1, proves the existence of a common fixed point for four self-maps under conditions that the map pairs satisfy a common property (EA) and are closed, JSR mappings satisfying an inequality involving the probabilistic distance functions.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
The document summarizes research on coupled fixed point theorems in ordered metric spaces. It presents definitions of mixed monotone mappings and coupled fixed points. It then proves a new coupled fixed point theorem (Theorem 8) for mappings satisfying a contractive condition using the concept of g-monotone mappings. The theorem establishes the existence of a coupled fixed point when the mappings satisfy inequality (1). Finally, it presents coupled coincidence and common fixed point results (Theorems 9) for mappings satisfying integral type contractions.
This document provides an introduction to symbolic math in MATLAB. It discusses differentiation and integration of functions using symbolic operators. Differentiation is defined as finding the rate of change of a function with respect to a variable. Integration finds the original function given its derivative. The document provides examples of differentiating and integrating simple functions in MATLAB's symbolic toolbox and exercises for the reader to practice.
The document presents a fixed point theorem for non-expansive mappings in 2-Banach spaces. It defines key concepts such as 2-norms, 2-Banach spaces, and non-expansive mappings. The main theorem proves that if mappings F and G satisfy certain conditions, including FG=G=I and a contraction-like inequality, then F and G have a common fixed point. The proof involves showing the sequence defined by an average of F and I converges to a fixed point shared by F and G. This generalized previous fixed point results for mappings on 2-Banach spaces.
This document contains instructions for Assignment 4 of the MA244 Analysis III module. It states that:
- 15% of the module credit comes from 4 assignments due on Mondays in weeks 4, 6, 8, and 10. Each assignment will be marked out of 25 based on answers to 3 randomly chosen 'B' questions and 1 'A' question.
- All questions for the assignment are due by December 7th, 2015 at 3pm and should include the student's name, department, and supervisor/TA. Work should be submitted in the appropriate location based on whether the student is in Maths or another department.
- The assignment contains 10 questions ranging from 0.1 to 4, with
8 fixed point theorem in complete fuzzy metric space 8 megha shrivastavaBIOLOGICAL FORUM
ABSTRACT: In this paper our works establish a new fixed point theorem for a different type of mapping in complete fuzzy metric space. Here we define a mapping by using some proved results and obtain a result on the actuality of fixed points. We inspired by the concept of Hossein Piri and Poom Kumam [15]. They introduced the fixed point theorem for generalized F-suzuki -contraction mappings in complete b-metric space. Next Robert plebaniak [16] express his idea by result “New generalized fuzzy metric space and fixed point theorem in fuzzy metric space”. This paper also induces comparing of the outcome with existing result in the literature.
Keywords: Fuzzy set, Fuzzy metric space, Cauchy sequence Non- decreasing sequence, Fixed point, Mapping.
This document provides an introduction and overview of Sylow's theorem regarding the construction of finite groups with specific numbers of Sylow p-subgroups. It begins with prerequisites and definitions, then presents three theorems:
Theorem 1 proves the existence of a group with qe Sylow p-subgroups for any e in a set E. Corollary 1 extends this to allow constructing groups with qem Sylow p-subgroups for any m. Theorem 2 addresses the special case of 2-subgroups, showing there exists a group with n Sylow 2-subgroups for any odd positive integer n. The document establishes notation and provides proofs of lemmas supporting each theorem. It aims to provide intuition on constructing groups to
This document defines functions and related terminology. It discusses:
- The definition of a function, domain, codomain, range, and related terms
- Properties of functions including one-to-one, onto, and bijective functions
- Inverse functions and how they relate to bijective functions
- Examples are provided to illustrate injective, surjective and bijective functions as well as calculating inverse functions.
This document contains answers to multiple questions about image processing concepts. For question 22a, the kernel formed by the outer product of vectors v and wT is determined to be separable. For question 22b, it is explained that a separable kernel w can be decomposed into two simpler kernels w1 and w2 such that w = w1 * w2. This allows the convolution to be computed more efficiently in two steps by first convolving w1 with the image and then convolving the result with w2, requiring fewer operations than a direct convolution with w.
1) The document discusses solving partial differential equations using variable separation. It presents the one-dimensional wave equation and solves it using variable separation.
2) It derives three cases for the solution based on whether k is positive, negative, or zero. It then presents the general solution as a summation involving sines and cosines.
3) It applies the general solution to two example problems of a vibrating string, finding the displacement as a function of position and time by satisfying the boundary conditions.
The document describes the RSA public-key cryptosystem. It discusses how RSA uses a public key for encryption and a private key for decryption. It also outlines the key steps for RSA, including key generation where two large prime numbers are used to calculate the public and private keys, encryption using the public key, and decryption using the private key. An example is provided to illustrate encrypting and decrypting a message using RSA.
INDUCTIVE LEARNING OF COMPLEX FUZZY RELATIONijcseit
The objective of this paper to investigate the notion of complex fuzzy set in general view. In constraint to a
traditional fuzzy set, the membership function of the complex fuzzy set, the range from [0.1] extended to a
unit circle in the complex plane. In this article the comprehensive mathematical operations on the complex
fuzzy set are presented. The basic operation of complex fuzzy set such as union, intersection, complement
of complex fuzzy set and complex fuzzy relation are studied. Also vector aggregation and fuzzy relation
over the complex fuzzy set are discussed. Two novel operations of complement and projection for complex
fuzzy relation are introduced.
This document presents two integrals involving the product of a generalized Riemann zeta function and an H-function. The H-function is a generalization of Fox's H-function that is defined through a Mellin-Barnes contour integral. Two integrals are derived: the first integral involves parameters m, n, p, q and the second integral involves parameters m, n, g, p, q. The integrals are proved by expressing the H-function as a Mellin-Barnes contour integral and using properties of the generalized Riemann zeta function and Gamma function. Special cases are discussed where the H-function reduces to other useful functions like Fox's H-function or a generalized Wright hypergeometric function
The document discusses relations and functions. It defines relations as subsets of Cartesian products of sets and describes how to classify relations as reflexive, symmetric, transitive, or an equivalence relation. It also defines functions, including their domain, codomain, and range. It describes how to classify functions as injective, surjective, or bijective. Examples are provided to illustrate these concepts of relations and functions.
Let Pn(x) be the Legendre polynomial of degree n. Then the generating function for Pn(x) is given by:
∞
1
Pn(x)tn = √
n=0
1 − 2xt + t2
Differentiating both sides with respect to t, we get:
∞
∑nPn(x)tn-1 = -xt(1 − 2xt + t2)-1/2 + (1 − 2xt + t2)-3/2
n=1
Multiplying both sides by (1 − 2xt + t2)1/2, we get:
∞
∑
This document discusses the chain rule for functions of multiple variables. It begins by reviewing the chain rule for single-variable functions, then extends it to functions of more variables. The chain rule is presented for cases where the dependent variable z is a function of intermediate variables x and y, which are themselves functions of independent variables s and t. General formulas are given using partial derivatives. Examples are worked out, such as finding the derivative of a function defined implicitly by an equation. Diagrams are used to illustrate the relationships between variables.
This document discusses relations and functions for class 12. It defines concepts like cartesian product, relations, domain and codomain, range, types of relations such as reflexive, symmetric, transitive and equivalence relations. It also defines functions, one-to-one and onto functions, bijective functions, composition of functions, and inverse functions. An example is provided to show that a given relation is reflexive but not symmetric or transitive. Another example shows a given function is one-to-one but not onto.
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
1) The document discusses inverse trigonometric functions such as sin-1, cos-1, tan-1, cot-1, sec-1, and cosec-1.
2) It explains that the inverse functions are defined by restricting the domains of the original trigonometric functions (sin, cos, etc.) so that they become one-to-one mappings.
3) Graphs of the inverse functions are obtained by reflecting the graphs of the original functions about the line y=x.
This document defines functions and relations. It discusses identifying the domain and range of functions and relations, evaluating functions, and performing operations on functions such as addition, subtraction, multiplication, division, and composition. It also covers graphing functions, including piecewise functions, absolute value functions, greatest and least integer functions. Key examples are provided to illustrate how to identify domains and ranges, evaluate functions, perform operations on functions, and graph different types of functions.
It turns out that an equation x^p+y^q=z^w has a solution in Z+ if and only if at least one exponent p or q or w equals 2.DOI:10.13140/RG.2.2.15459.22567/7
Fixed point result in menger space with ea propertyAlexander Decker
This document presents a fixed point theorem for four self-maps in a Menger space. It begins by defining key concepts related to Menger spaces including probabilistic metric spaces, t-norms, neighborhoods, convergence, Cauchy sequences, and completeness. It then introduces properties like weakly compatible maps, property (EA), and JSR mappings. The main result, Theorem 3.1, proves the existence of a common fixed point for four self-maps under conditions that the map pairs satisfy a common property (EA) and are closed, JSR mappings satisfying an inequality involving the probabilistic distance functions.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
The document summarizes research on coupled fixed point theorems in ordered metric spaces. It presents definitions of mixed monotone mappings and coupled fixed points. It then proves a new coupled fixed point theorem (Theorem 8) for mappings satisfying a contractive condition using the concept of g-monotone mappings. The theorem establishes the existence of a coupled fixed point when the mappings satisfy inequality (1). Finally, it presents coupled coincidence and common fixed point results (Theorems 9) for mappings satisfying integral type contractions.
This document provides an introduction to symbolic math in MATLAB. It discusses differentiation and integration of functions using symbolic operators. Differentiation is defined as finding the rate of change of a function with respect to a variable. Integration finds the original function given its derivative. The document provides examples of differentiating and integrating simple functions in MATLAB's symbolic toolbox and exercises for the reader to practice.
The document presents a fixed point theorem for non-expansive mappings in 2-Banach spaces. It defines key concepts such as 2-norms, 2-Banach spaces, and non-expansive mappings. The main theorem proves that if mappings F and G satisfy certain conditions, including FG=G=I and a contraction-like inequality, then F and G have a common fixed point. The proof involves showing the sequence defined by an average of F and I converges to a fixed point shared by F and G. This generalized previous fixed point results for mappings on 2-Banach spaces.
This document contains instructions for Assignment 4 of the MA244 Analysis III module. It states that:
- 15% of the module credit comes from 4 assignments due on Mondays in weeks 4, 6, 8, and 10. Each assignment will be marked out of 25 based on answers to 3 randomly chosen 'B' questions and 1 'A' question.
- All questions for the assignment are due by December 7th, 2015 at 3pm and should include the student's name, department, and supervisor/TA. Work should be submitted in the appropriate location based on whether the student is in Maths or another department.
- The assignment contains 10 questions ranging from 0.1 to 4, with
8 fixed point theorem in complete fuzzy metric space 8 megha shrivastavaBIOLOGICAL FORUM
ABSTRACT: In this paper our works establish a new fixed point theorem for a different type of mapping in complete fuzzy metric space. Here we define a mapping by using some proved results and obtain a result on the actuality of fixed points. We inspired by the concept of Hossein Piri and Poom Kumam [15]. They introduced the fixed point theorem for generalized F-suzuki -contraction mappings in complete b-metric space. Next Robert plebaniak [16] express his idea by result “New generalized fuzzy metric space and fixed point theorem in fuzzy metric space”. This paper also induces comparing of the outcome with existing result in the literature.
Keywords: Fuzzy set, Fuzzy metric space, Cauchy sequence Non- decreasing sequence, Fixed point, Mapping.
This document provides an introduction and overview of Sylow's theorem regarding the construction of finite groups with specific numbers of Sylow p-subgroups. It begins with prerequisites and definitions, then presents three theorems:
Theorem 1 proves the existence of a group with qe Sylow p-subgroups for any e in a set E. Corollary 1 extends this to allow constructing groups with qem Sylow p-subgroups for any m. Theorem 2 addresses the special case of 2-subgroups, showing there exists a group with n Sylow 2-subgroups for any odd positive integer n. The document establishes notation and provides proofs of lemmas supporting each theorem. It aims to provide intuition on constructing groups to
This document defines functions and related terminology. It discusses:
- The definition of a function, domain, codomain, range, and related terms
- Properties of functions including one-to-one, onto, and bijective functions
- Inverse functions and how they relate to bijective functions
- Examples are provided to illustrate injective, surjective and bijective functions as well as calculating inverse functions.
This document contains answers to multiple questions about image processing concepts. For question 22a, the kernel formed by the outer product of vectors v and wT is determined to be separable. For question 22b, it is explained that a separable kernel w can be decomposed into two simpler kernels w1 and w2 such that w = w1 * w2. This allows the convolution to be computed more efficiently in two steps by first convolving w1 with the image and then convolving the result with w2, requiring fewer operations than a direct convolution with w.
1) The document discusses solving partial differential equations using variable separation. It presents the one-dimensional wave equation and solves it using variable separation.
2) It derives three cases for the solution based on whether k is positive, negative, or zero. It then presents the general solution as a summation involving sines and cosines.
3) It applies the general solution to two example problems of a vibrating string, finding the displacement as a function of position and time by satisfying the boundary conditions.
The document describes the RSA public-key cryptosystem. It discusses how RSA uses a public key for encryption and a private key for decryption. It also outlines the key steps for RSA, including key generation where two large prime numbers are used to calculate the public and private keys, encryption using the public key, and decryption using the private key. An example is provided to illustrate encrypting and decrypting a message using RSA.
INDUCTIVE LEARNING OF COMPLEX FUZZY RELATIONijcseit
The objective of this paper to investigate the notion of complex fuzzy set in general view. In constraint to a
traditional fuzzy set, the membership function of the complex fuzzy set, the range from [0.1] extended to a
unit circle in the complex plane. In this article the comprehensive mathematical operations on the complex
fuzzy set are presented. The basic operation of complex fuzzy set such as union, intersection, complement
of complex fuzzy set and complex fuzzy relation are studied. Also vector aggregation and fuzzy relation
over the complex fuzzy set are discussed. Two novel operations of complement and projection for complex
fuzzy relation are introduced.
This document presents two integrals involving the product of a generalized Riemann zeta function and an H-function. The H-function is a generalization of Fox's H-function that is defined through a Mellin-Barnes contour integral. Two integrals are derived: the first integral involves parameters m, n, p, q and the second integral involves parameters m, n, g, p, q. The integrals are proved by expressing the H-function as a Mellin-Barnes contour integral and using properties of the generalized Riemann zeta function and Gamma function. Special cases are discussed where the H-function reduces to other useful functions like Fox's H-function or a generalized Wright hypergeometric function
The document discusses relations and functions. It defines relations as subsets of Cartesian products of sets and describes how to classify relations as reflexive, symmetric, transitive, or an equivalence relation. It also defines functions, including their domain, codomain, and range. It describes how to classify functions as injective, surjective, or bijective. Examples are provided to illustrate these concepts of relations and functions.
Let Pn(x) be the Legendre polynomial of degree n. Then the generating function for Pn(x) is given by:
∞
1
Pn(x)tn = √
n=0
1 − 2xt + t2
Differentiating both sides with respect to t, we get:
∞
∑nPn(x)tn-1 = -xt(1 − 2xt + t2)-1/2 + (1 − 2xt + t2)-3/2
n=1
Multiplying both sides by (1 − 2xt + t2)1/2, we get:
∞
∑
This document discusses the chain rule for functions of multiple variables. It begins by reviewing the chain rule for single-variable functions, then extends it to functions of more variables. The chain rule is presented for cases where the dependent variable z is a function of intermediate variables x and y, which are themselves functions of independent variables s and t. General formulas are given using partial derivatives. Examples are worked out, such as finding the derivative of a function defined implicitly by an equation. Diagrams are used to illustrate the relationships between variables.
This document discusses relations and functions for class 12. It defines concepts like cartesian product, relations, domain and codomain, range, types of relations such as reflexive, symmetric, transitive and equivalence relations. It also defines functions, one-to-one and onto functions, bijective functions, composition of functions, and inverse functions. An example is provided to show that a given relation is reflexive but not symmetric or transitive. Another example shows a given function is one-to-one but not onto.
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
1) The document discusses inverse trigonometric functions such as sin-1, cos-1, tan-1, cot-1, sec-1, and cosec-1.
2) It explains that the inverse functions are defined by restricting the domains of the original trigonometric functions (sin, cos, etc.) so that they become one-to-one mappings.
3) Graphs of the inverse functions are obtained by reflecting the graphs of the original functions about the line y=x.
This document defines functions and relations. It discusses identifying the domain and range of functions and relations, evaluating functions, and performing operations on functions such as addition, subtraction, multiplication, division, and composition. It also covers graphing functions, including piecewise functions, absolute value functions, greatest and least integer functions. Key examples are provided to illustrate how to identify domains and ranges, evaluate functions, perform operations on functions, and graph different types of functions.
This document provides an overview of various types of functions and their graphs. It begins with linear functions of the form y=mx+c and discusses how shifting these functions along the x- or y-axis changes their graphs. It then covers quadratic, square root, cube, reciprocal, constant, identity and absolute value functions. Piecewise, polynomial, algebraic, and transcendental functions are also defined. The document discusses bounded vs unbounded functions and concludes by examining circular and hyperbolic functions and their graphs.
This document provides an introduction to functions and limits. It defines key concepts such as domain, range, and different types of functions including algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Examples are provided to illustrate how to find the domain and range of functions, evaluate functions, and draw graphs of functions. Function notation and the concept of a function as a rule that assigns each input to a single output are also explained.
This document provides an introduction to functions and their key concepts. It defines what a function is, using examples to illustrate functions that relate variables. Functions have a domain and range, and can be represented graphically. Common types of functions are discussed, including algebraic functions like polynomials and rational functions, as well as trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Methods for determining a function's domain and range and drawing its graph are presented.
Fourier series can be used to represent periodic functions or functions on an interval. They involve expanding the function as a sum of sines and cosines. Fourier series are useful because they can represent discontinuous functions, unlike Taylor series. When expanding a function over an interval, the Fourier series is only guaranteed to apply over that initial interval unless additional information is given about the periodicity of the function. Fourier series can be adapted to any periodic interval by changing the variables. As an example application, a square wave can be analyzed using its Fourier components via a Fourier series expansion.
The document discusses Cartesian products, domains, ranges, and co-domains of relations and functions through examples and definitions. It explains that the Cartesian product of sets A and B, written as A×B, is the set of all ordered pairs (a,b) where a is an element of A and b is an element of B. It also defines what constitutes a relation between two sets and provides examples of relations and functions, discussing their domains and ranges. Arrow diagrams are presented to illustrate various functions along with questions and their solutions related to relations and functions.
This document provides an introduction to functions and their key concepts. It defines a function as a rule that assigns each element in one set to a unique element in another set. Functions can be represented graphically and algebraically. Common types of functions discussed include polynomial, linear, constant, rational, trigonometric, inverse trigonometric, exponential, logarithmic, and hyperbolic functions. Examples are provided to illustrate domain, range, and graphing of different function types.
This document discusses different types of relations and functions. It defines equivalence relations, identity relations, empty relations, universal relations, one-to-one functions, onto functions, bijective functions, composition of functions, and invertible functions. It provides examples to illustrate these concepts.
The document discusses orthogonal polynomials, focusing on Legendre and Chebyshev polynomials. It introduces Hilbert spaces and self-adjoint operators, describing properties like Hermitian matrices having real eigenvalues. It defines the L2 space and shows how differential operators can act as self-adjoint. Legendre polynomials are defined using Rodrigue's formula and their generating function is explored. The first few Legendre polynomials are shown.
A Proof of Hypothesis Riemann and it is proven that apply only for equation ζ(z)=0.Also here it turns out that it does not apply in General Case.(DOI:10.13140/RG.2.2.10888.34563/5)
The document provides an overview of concepts in functional analysis that will be covered in a math camp, including: function spaces, metric spaces, dense subsets, linear spaces, linear functionals, norms, Euclidean spaces, orthogonality, separable spaces, complete metric spaces, Hilbert spaces, and convex functions. Examples are given for each concept to illustrate the definitions.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
The document discusses functions and relations. It defines functions, relations, and domain and range. It provides examples of expressing relations in set notation, tabular form, equations, graphs, and mappings. It also discusses evaluating, adding, multiplying, dividing, and composing functions. Graphs of various functions like absolute value, piecewise, greatest integer, and least integer functions are also explained.
Presentation on Fourier Series
contents are:-
Euler’s Formula
Functions having point of discontinuity
Change of interval
Even and Odd functions
Half Range series
Harmonic analysis
Signal Processing Introduction using Fourier TransformsArvind Devaraj
1) The document introduces signal processing by discussing signals, systems, and transforms. It defines signals as functions of time or space and systems as maps that manipulate signals. Transforms represent signals in different domains like frequency to simplify operations.
2) Signals can be represented in the frequency domain using Fourier transforms. This makes operations like filtering easier. Low frequencies represent overall shape while high frequencies are details like noise or edges.
3) Linear and time-invariant systems can be characterized by their impulse response. The output is the convolution of the input and impulse response. Convolution is a mechanism that shapes signals to produce outputs.
The document discusses inverse functions and logarithms. It begins by introducing the concept of an inverse function using an example of bacteria population growth over time. It then defines inverse functions formally and discusses their key properties. The document explains that a function must be one-to-one to have an inverse function. It introduces the natural logarithm as the inverse of the exponential function with base e and discusses properties of logarithmic functions like logarithmic laws. Graphs of exponential, logarithmic and natural logarithmic functions are presented.
The document discusses quantum mechanical concepts including:
1) The time derivative of the momentum expectation value satisfies an equation involving the potential gradient.
2) For an infinite potential well, the kinetic energy expectation value is proportional to n^2/a^2 and the potential energy expectation value vanishes.
3) Eigenfunctions of an eigenvalue problem under certain boundary conditions correspond to positive eigenvalues that are sums of squares of integer multiples of pi.
A function transforms inputs (domain) to outputs (range). The inverse of a function reverses the inputs and outputs, so the domain of the inverse is the original function's range, and the range of the inverse is the original function's domain. To have an inverse, a function must be one-to-one, meaning each input maps to a single unique output. The graph of an inverse is a reflection of the original function's graph across the line y=x.
Exercise 5.6.28. For each of the following descriptions of a function.pdfpetercoiffeur18
Exercise 5.6.28. For each of the following descriptions of a function f, find a suh S of the
domain of f such that the restriction fIs is a bijection onto the range of f. (a)f : N N, wheref(n)-n
+ 1 if n is odd and/(n) =n/2 if n is even (b)f : R Z, wheref(x)- . Here lxl denotes the floor\' or.
(c)f : R R, where,f(x) =x3-3x2 , (d)f : R R, where,f(x) =x4-4x2 (e)f : R R, wheref(x)=xe-x. ,
find a subset 7 The floor function from R to Z assigns to each x R the greatest integer less than or
equal to x. It is denoted by Lx. For example, L = 3 and L-r=-4 ie, | | = 3
Solution
A bijective function is a one to one function that is form every x value there will be exactly one y
value.
In simopler terms each element in the domain set of a function when plugged in the function
gives a distinct value of the function
that is a part of the range of the function.
in our problem we need to find the subset of a domain for which the function is bijective
a> f : N-->N , f(n) = n+1
the domain and the range are all natural numbers
that is n E [0,1,2,3,4,.....N]
if n is odd
f(n)= n+1 ,this is a straight line and this function will be bijective for all odd natural numbers that
is n E (2k+1) will be the
domain in which f(n) will be bijective ,as for all odd values of n we\'ll get distinct f values.
n E [1 , 2k+1] , here k = 1,2,3........k just the integral vaslues
when n is even
f(n) = n/2 , this again is a straight line so the domain in whicj f(n) is bijective will be all even
natural, numbers\\that is
n E 2k ,k E {1,2,3,4...N}
b> f(x) = [x]
x : Z-->Z
here f(x) represents a step function
and a step function have the same range for all the x values within a particular step domain
so f(x) = [x] , cannot be a one to one function as we do not get distinct f(x) values for different x
values
c> f(x) = x^3-3x^2
f(x)= x^2(x-3)
this function will be a curve with zeros at x = 0 and 3
and it will not be a bijective or one to one function over the entire real number line.
d>
f(x) = x^4-4x^2
this function will be a curve with zeros at x = 0 and 2 and -2
and it will not be a bijective or one to one function over the entire real number line..
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1. Mathematics, in general, is fundamentally the science of
self-evident things. — FELIX KLEIN
2.1 Introduction
In Chapter 1, we have studied that the inverse of a function
f, denoted by f –1
, exists if f is one-one and onto. There are
many functions which are not one-one, onto or both and
hence we can not talk of their inverses. In Class XI, we
studied that trigonometric functions are not one-one and
onto over their natural domains and ranges and hence their
inverses do not exist. In this chapter, we shall study about
the restrictions on domains and ranges of trigonometric
functions which ensure the existence of their inverses and
observe their behaviour through graphical representations.
Besides, some elementary properties will also be discussed.
The inverse trigonometric functions play an important
role in calculus for they serve to define many integrals.
The concepts of inverse trigonometric functions is also used in science and engineering.
2.2 Basic Concepts
In Class XI, we have studied trigonometric functions, which are defined as follows:
sine function, i.e., sine : R → [– 1, 1]
cosine function, i.e., cos : R → [– 1, 1]
tangent function, i.e., tan : R – { x : x = (2n + 1)
2
π
, n ∈ Z} → R
cotangent function, i.e., cot : R – { x : x = nπ, n ∈ Z} → R
secant function, i.e., sec : R – { x : x = (2n + 1)
2
π
, n ∈ Z} → R – (– 1, 1)
cosecant function, i.e., cosec : R – { x : x = nπ, n ∈ Z} → R – (– 1, 1)
Chapter 2
INVERSE TRIGONOMETRIC
FUNCTIONS
Arya Bhatta
(476-550A. D.)
2. 34 MATHEMATICS
We have also learnt in Chapter 1 that if f : X→Ysuch that f(x) = y is one-one and
onto, then we can define a unique function g : Y→X such that g(y) = x, where x ∈ X
and y = f (x), y ∈ Y. Here, the domain of g = range of f and the range of g = domain
of f. The function g is called the inverse of f and is denoted by f –1
. Further, g is also
one-one and onto and inverse of g is f. Thus, g–1
= (f –1
)–1
= f. We also have
(f –1
o f ) (x) = f –1
(f (x)) = f –1
(y) = x
and (f o f –1
) (y) = f (f –1
(y)) = f (x) = y
Since the domain of sine function is the set of all real numbers and range is the
closed interval [–1, 1]. If we restrict its domain to ,
2 2
−π π
, then it becomes one-one
and onto with range [– 1, 1]. Actually, sine function restricted to any of the intervals
3 –
,
2 2
− π π
, ,
2 2
−π π
,
3
,
2 2
π π
etc., is one-one and its range is [–1, 1]. We can,
therefore, define the inverse of sine function in each of these intervals. We denote the
inverse of sine function by sin–1
(arc sine function). Thus, sin–1
is a function whose
domain is [– 1, 1] and range could be any of the intervals
3
,
2 2
− π −π
, ,
2 2
−π π
or
3
,
2 2
π π
, and so on. Corresponding to each such interval, we get a branch of the
function sin–1. The branch with range ,
2 2
−π π
is called the principal value branch,
whereas other intervals as range give different branches of sin–1
. When we refer
to the function sin–1
, we take it as the function whose domain is [–1, 1] and range is
,
2 2
−π π
. We write sin–1 : [–1, 1] → ,
2 2
−π π
From the definition of the inverse functions, it follows that sin (sin–1
x) = x
if – 1 ≤ x ≤ 1 and sin–1 (sin x) = x if
2 2
x
π π
− ≤ ≤ . In other words, if y = sin–1 x, then
sin y = x.
Remarks
(i) We know from Chapter 1, that if y =f (x) is an invertible function, then x= f –1 (y).
Thus, the graph of sin–1 function can be obtained from the graph of original
function by interchanging x and y axes, i.e., if (a, b) is a point on the graph of
sine function, then (b,a) becomes the corresponding point on the graph of inverse
3. INVERSE TRIGONOMETRIC FUNCTIONS 35
of sine function. Thus, the graph of the function y = sin–1
x can be obtained from
the graph of y = sin x by interchanging x and yaxes. The graphs of y = sin x and
y = sin–1
x are as given in Fig 2.1 (i), (ii), (iii). The dark portion of the graph of
y = sin–1
x represent the principal value branch.
(ii) It can be shown that the graph of an inverse function can be obtained from the
corresponding graph of original function as a mirror image (i.e., reflection) along
the line y = x. This can be visualised by looking the graphs of y = sin x and
y = sin–1 x as given in the same axes (Fig 2.1 (iii)).
Like sine function, the cosine function is a function whose domain is the set of all
real numbers and range is the set [–1, 1]. If we restrict the domain of cosine function
to [0, π], then it becomes one-one and onto with range [–1, 1]. Actually, cosine function
Fig 2.1 (ii) Fig 2.1 (iii)
Fig 2.1 (i)
4. 36 MATHEMATICS
restricted to any of the intervals [– π, 0], [0,π], [π, 2π] etc., is bijective with range as
[–1, 1]. We can, therefore, define the inverse of cosine function in each of these
intervals. We denote the inverse of the cosine function by cos–1
(arc cosine function).
Thus, cos–1
is a function whose domain is [–1, 1] and range
could be any of the intervals [–π, 0], [0, π], [π, 2π] etc.
Corresponding to each such interval, we get a branch of the
function cos–1
. The branch with range [0, π] is calledthe principal
value branch of the function cos–1
. We write
cos–1
: [–1, 1] → [0, π].
The graph of the function given by y = cos–1
x can be drawn
in the same way as discussed about the graph of y = sin–1
x. The
graphs of y = cos x and y = cos–1
xare given in Fig 2.2 (i)and (ii).
Fig 2.2 (ii)
Let us now discuss cosec–1x and sec–1x as follows:
Since, cosec x =
1
sin x
, the domain of the cosec function is the set {x : x ∈ R and
x ≠ nπ, n ∈ Z} and the range is the set {y : y ∈ R, y ≥ 1 or y ≤ –1} i.e., the set
R – (–1, 1). It means that y = cosec x assumes all real values except –1 < y < 1 and is
not defined for integral multiple of π. If we restrict the domain of cosec function to
,
2 2
π π
−
–{0},thenitisonetooneand onto with itsrange as theset R – (– 1,1).Actually,
cosec function restricted to any of the intervals
3
, { }
2 2
− π −π
− −π
, ,
2 2
−π π
– {0},
3
, { }
2 2
π π
− π
etc., is bijective and its range is the set of all real numbers R – (–1, 1).
Fig 2.2 (i)
5. INVERSE TRIGONOMETRIC FUNCTIONS 37
Thus cosec–1
can be defined as a function whose domain is R – (–1, 1) and range could
be any of the intervals , {0}
2 2
−π π
−
,
3
, { }
2 2
− π −π
− −π
,
3
, { }
2 2
π π
− π
etc. The
function corresponding to the range , {0}
2 2
−π π
−
is called the principal value branch
of cosec–1. We thus have principal branch as
cosec–1
: R – (–1, 1) → , {0}
2 2
−π π
−
The graphs of y = cosec x and y = cosec–1
x are given in Fig 2.3 (i), (ii).
Also, since sec x =
1
cos x
, the domain of y = sec x is the setR – {x : x= (2n + 1)
2
π
,
n ∈ Z} and range is the set R – (–1, 1). It means that sec (secant function) assumes
all real values except –1 < y < 1 and is not defined for odd multiples of
2
π
. If we
restrict the domain of secant function to [0, π] – {
2
π
}, then it is one-one and onto with
Fig 2.3 (i) Fig 2.3 (ii)
6. 38 MATHEMATICS
its range as the set R – (–1, 1). Actually, secant function restricted to any of the
intervals [–π, 0] – {
2
−π
}, [0, ] –
2
π
π
, [π, 2π] – {
3
2
π
} etc., is bijective and its range
is R – {–1, 1}. Thus sec–1
can be defined as a function whose domain is R– (–1, 1) and
range could be any of the intervals [– π, 0] – {
2
−π
}, [0, π] – {
2
π
}, [π, 2π] – {
3
2
π
} etc.
Corresponding to each of these intervals, we get different branches of the function sec–1.
The branch with range [0, π] – {
2
π
} is called the principal value branch of the
function sec–1. We thus have
sec–1
: R – (–1,1) → [0, π] – {
2
π
}
The graphs of the functions y = sec x and y = sec-1
x are given in Fig 2.4 (i), (ii).
Finally, we now discuss tan–1
and cot–1
We know that the domain of the tan function (tangent function) is the set
{x : x ∈ R and x ≠ (2n +1)
2
π
, n ∈ Z} and the range is R. It means that tan function
is not defined for odd multiples of
2
π
. If we restrict the domain of tangent function to
Fig 2.4 (i) Fig 2.4 (ii)
7. INVERSE TRIGONOMETRIC FUNCTIONS 39
,
2 2
−π π
, then it is one-one and onto with its range as R. Actually, tangent function
restricted to any of the intervals
3
,
2 2
− π −π
, ,
2 2
−π π
,
3
,
2 2
π π
etc., is bijective
and its range is R. Thus tan–1
can be defined as a function whose domain is R and
range could be any of the intervals
3
,
2 2
− π −π
, ,
2 2
−π π
,
3
,
2 2
π π
and so on. These
intervals give different branches of the function tan–1
. The branch with range ,
2 2
−π π
is called the principal value branch of the function tan–1
.
We thus have
tan–1
: R → ,
2 2
−π π
The graphs of the function y = tan x and y = tan–1
x are given in Fig 2.5 (i), (ii).
Fig 2.5 (i) Fig 2.5 (ii)
We know that domain of the cot function (cotangent function) is the set
{x : x ∈ R and x ≠ nπ, n ∈ Z} and range is R. It means that cotangent function is not
defined for integral multiples of π. If we restrict the domain of cotangent function to
(0, π), then it is bijective with and its range as R. In fact, cotangent function restricted
to any of the intervals (–π, 0), (0, π), (π, 2π) etc., is bijective and its range is R. Thus
cot –1
can be defined as a function whose domain is the R and range as any of the
8. 40 MATHEMATICS
intervals (–π, 0), (0, π), (π, 2π) etc. These intervals give different branches of the
function cot–1
. The function with range (0, π) is called the principal value branch of
the function cot–1. We thus have
cot–1 : R → (0, π)
The graphs of y = cot x and y = cot–1
x are given in Fig 2.6 (i), (ii).
Fig 2.6 (i) Fig 2.6 (ii)
The following table gives the inverse trigonometric function (principal value
branches) along with their domains and ranges.
sin–1
: [–1, 1] → ,
2 2
π π
−
cos–1 : [–1, 1] → [0, π]
cosec–1
: R – (–1,1) → ,
2 2
π π
−
– {0}
sec–1
: R – (–1, 1) → [0, π] – { }
2
π
tan–1
: R → ,
2 2
−π π
cot–1 : R → (0, π)
9. INVERSE TRIGONOMETRIC FUNCTIONS 41
Note
1. sin–1
x should not be confused with (sin x)–1
. In fact (sin x)–1
=
1
sin x
and
similarly for other trigonometric functions.
2. Whenever no branch of an inverse trigonometric functions is mentioned, we
mean the principal value branch of that function.
3. The value of an inverse trigonometric functions which lies in the range of
principal branch is called the principal value of that inverse trigonometric
functions.
We now consider some examples:
Example 1 Find the principal value of sin–1
1
2
.
Solution Let sin–1
1
2
= y. Then, sin y =
1
2
.
We know that the range of the principal value branch of sin–1
is ,
2 2
−π π
and
sin
4
π
=
1
2
. Therefore, principal value of sin–1
1
2
is
4
π
Example 2 Find the principal value of cot–1 1
3
−
Solution Let cot–1
1
3
−
= y. Then,
1
cot cot
33
y
− π
= = −
= cot
3
π
π −
=
2
cot
3
π
We know that the range of principal value branch of cot–1
is (0, π) and
cot
2
3
π
=
1
3
−
. Hence, principal value of cot–1
1
3
−
is
2
3
π
EXERCISE 2.1
Find the principal values of the following:
1. sin–1
1
2
−
2. cos–1
3
2
3. cosec–1
(2)
4. tan–1
( 3)− 5. cos–1
1
2
−
6. tan–1
(–1)
10. 42 MATHEMATICS
7. sec–1
2
3
8. cot–1
( 3) 9. cos–1
1
2
−
10. cosec–1
( 2− )
Find the values of the following:
11. tan–1 (1) + cos–1
1
2
−
+ sin–1
1
2
−
12. cos–1
1
2
+ 2 sin–1
1
2
13. If sin–1 x = y, then
(A) 0 ≤ y ≤ π (B)
2 2
y
π π
− ≤ ≤
(C) 0 < y < π (D)
2 2
y
π π
− < <
14. tan–1
( )1
3 sec 2−
− − is equal to
(A) π (B)
3
π
− (C)
3
π
(D)
2
3
π
2.3 Properties of InverseTrigonometric Functions
In this section, we shall prove some important properties of inverse trigonometric
functions. It may be mentioned here that these results are valid within the principal
value branches of the corresponding inverse trigonometric functions and wherever
they are defined. Some results may not be valid for all values of the domains of inverse
trigonometric functions. In fact, they will be valid only for some values of x for which
inverse trigonometric functions are defined. We will not go into the details of these
values of x in the domain as this discussion goes beyond the scope of this text book.
Let us recall that if y = sin–1
x, then x = sin y and if x = sin y, then y= sin–1
x. This is
equivalent to
sin (sin–1
x) = x, x ∈ [– 1, 1] and sin–1
(sin x) = x, x ∈ ,
2 2
π π
−
Same is true for other five inverse trigonometric functions as well. We now prove
some properties of inverse trigonometric functions.
1. (i) sin–1 1
x
= cosec–1
x, x ≥≥≥≥≥ 1 or x ≤≤≤≤≤ – 1
(ii) cos–1 1
x
= sec–1
x, x ≥≥≥≥≥ 1 or x ≤≤≤≤≤ – 1
11. INVERSE TRIGONOMETRIC FUNCTIONS 43
(iii) tan–1 1
x
= cot–1
x, x > 0
To prove the first result, we put cosec–1 x = y, i.e., x = cosec y
Therefore
1
x
= sin y
Hence sin–1
1
x
= y
or sin–1
1
x
= cosec–1
x
Similarly, we can prove the other parts.
2. (i) sin–1
(–x) = – sin–1
x, x ∈∈∈∈∈ [– 1, 1]
(ii) tan–1
(–x) = – tan–1
x, x ∈∈∈∈∈ R
(iii) cosec–1 (–x) = – cosec–1 x, |x | ≥≥≥≥≥ 1
Let sin–1 (–x) = y, i.e., –x = sin y so that x = – sin y, i.e., x = sin (–y).
Hence sin–1 x = – y = – sin–1 (–x)
Therefore sin–1
(–x) = – sin–1
x
Similarly, we can prove the other parts.
3. (i) cos–1
(–x) = πππππ – cos–1
x, x ∈∈∈∈∈ [– 1, 1]
(ii) sec–1
(–x) = πππππ – sec–1
x, |x| ≥≥≥≥≥ 1
(iii) cot–1
(–x) = πππππ – cot–1
x, x ∈∈∈∈∈ R
Let cos–1
(–x) = y i.e., – x = cos y so that x = – cos y = cos (π – y)
Therefore cos–1 x = π – y = π – cos–1 (–x)
Hence cos–1 (–x) = π – cos–1 x
Similarly, we can prove the other parts.
4. (i) sin–1 x + cos–1 x =
2
π
, x ∈∈∈∈∈ [– 1, 1]
(ii) tan–1
x + cot–1
x =
2
π
, x ∈∈∈∈∈ R
(iii) cosec–1
x + sec–1
x =
2
π
, |x | ≥≥≥≥≥ 1
Let sin–1
x = y. Then x = sin y = cos
2
y
π
−
Therefore cos–1
x =
2
y
π
− =
–1
sin
2
x
π
−
12. 44 MATHEMATICS
Hence sin–1 x + cos–1 x =
2
π
Similarly, we can prove the other parts.
5. (i) tan–1
x + tan–1
y = tan–1
1
x + y
– xy
, xy < 1
(ii) tan–1 x – tan–1 y = tan–1
1
x – y
+ xy
, xy > – 1
Let tan–1
x = θ and tan–1
y = φ. Then x = tan θ, y = tan φ
Now
tan tan
tan( )
1 tan tan 1
x y
xy
θ+ φ +
θ+φ = =
− θ φ −
This gives θ + φ = tan–1
1
x y
xy
+
−
Hence tan–1
x + tan–1
y = tan–1
1
x y
xy
+
−
In the above result, if we replace y by –y, we get the second result and by replacing
y by x, we get the third result as given below.
6. (i) 2tan–1
x = sin–1
2
2
1+
x
x
, |x| ≤≤≤≤≤ 1
(ii) 2tan–1
x = cos–1
2
2
1 –
1+
x
x
, x ≥≥≥≥≥ 0
(iii) 2 tan–1
x = tan–1
2
2
1 –
x
x
, – 1 < x < 1
Let tan–1
x = y, then x = tan y. Now
sin–1
2
2
1
x
x+
= sin–1
2
2tan
1 tan
y
y+
= sin–1
(sin 2y) = 2y = 2tan–1
x
13. INVERSE TRIGONOMETRIC FUNCTIONS 45
Also cos–1
2
2
1
1
x
x
−
+
= cos–1
2
2
1 tan
1 tan
y
y
−
+
= cos–1
(cos 2y) = 2y = 2tan–1
x
(iii) Can be worked out similarly.
We now consider some examples.
Example 3 Show that
(i) sin–1
( )2
2 1x x− = 2 sin–1
x,
1 1
2 2
x− ≤ ≤
(ii) sin–1
( )2
2 1x x− = 2 cos–1
x,
1
1
2
x≤ ≤
Solution
(i) Let x = sin θ. Then sin–1
x = θ. We have
sin–1
( )2
2 1x x− = sin–1
( )2
2sin 1 sinθ − θ
= sin–1 (2sinθ cosθ) = sin–1 (sin2θ) = 2θ
= 2 sin–1
x
(ii) Take x = cos θ, then proceeding as above, we get, sin–1
( )2
2 1x x− = 2 cos–1
x
Example 4 Show that tan–1 –1 –11 2 3
tan tan
2 11 4
+ =
Solution By property 5 (i), we have
L.H.S. =
–1 –11 2
tan tan
2 11
+ –1 1
1 2
152 11tan tan
1 2 201
2 11
−
+
= =
− ×
=
1 3
tan
4
−
= R.H.S.
Example 5 Express 1 cos
tan
1 sin
x
x
−
−
,
3
2 2
π π−
< <x in the simplest form.
Solution We write
2 2
1 –1
2 2
cos sin
cos 2 2tan tan
1 sin cos sin 2sin cos
2 2 2 2
x x
x
x x x xx
−
−
=
− + −
14. 46 MATHEMATICS
=
–1
2
cos sin cos sin
2 2 2 2
tan
cos sin
2 2
x x x x
x x
+ −
−
=
–1
cos sin
2 2tan
cos sin
2 2
x x
x x
+
−
–1
1 tan
2tan
1 tan
2
x
x
+
=
−
=
–1
tan tan
4 2 4 2
x x π π
+ = +
Alternatively,
–1 –1 –1
2
sin sin
cos 2 2
tan tan tan
21 sin
1 cos 1 cos
2 2
x
x
x
xx
x
π π −
− = = π π −− − − −
=
–1
2
2 2
2sin cos
4 4
tan
2
2sin
4
x x
x
π − π −
π −
=
–1 2
tan cot
4
x π −
–1 2
tan tan
2 4
x π π −
= −
=
–1
tan tan
4 2
x π
+
4 2
xπ
= +
Example 6 Write
–1
2
1
cot
1x
−
, x > 1 in the simplest form.
Solution Let x = sec θ, then 2
1x − = 2
sec 1 tanθ − = θ
15. INVERSE TRIGONOMETRIC FUNCTIONS 47
Therefore,
–1
2
1
cot
1x −
= cot–1
(cot θ) = θ = sec–1
x, which is the simplest form.
Example 7 Prove that tan–1 x +
–1
2
2
tan
1
x
x−
= tan–1
3
2
3
1 3
x x
x
−
−
,
1
| |
3
x <
Solution Let x = tan θ. Then θ = tan–1
x. We have
R.H.S. =
3 3
–1 –1
2 2
3 3tan tan
tan tan
1 3 1 3tan
x x
x
− θ− θ
=
− − θ
= tan–1
(tan3θ) = 3θ = 3tan–1
x = tan–1
x + 2 tan–1
x
= tan–1 x + tan–1
2
2
1
x
x−
= L.H.S. (Why?)
Example 8 Find the value of cos (sec–1 x + cosec–1 x), | x | ≥ 1
Solution We have cos (sec–1
x + cosec–1
x) = cos
2
π
= 0
EXERCISE 2.2
Prove the following:
1. 3sin–1
x = sin–1
(3x – 4x3
),
1 1
– ,
2 2
x
∈
2. 3cos–1 x = cos–1 (4x3 – 3x),
1
, 1
2
x
∈
3. tan–1 1 12 7 1
tan tan
11 24 2
− −
+ =
4.
1 1 11 1 31
2 tan tan tan
2 7 17
− − −
+ =
Write the following functions in the simplest form:
5.
2
1 1 1
tan
x
x
− + −
, x ≠ 0 6.
1
2
1
tan
1x
−
−
, |x | > 1
7.
1 1 cos
tan
1 cos
x
x
−
−
+
, 0 < x < π 8. 1 cos sin
tan
cos sin
x x
x x
− −
+
, 0 < x< π
16. 48 MATHEMATICS
9.
1
2 2
tan
x
a x
−
−
, |x | < a
10.
2 3
1
3 2
3
tan
3
a x x
a ax
− −
−
, a > 0;
3 3
−
< <
a a
x
Find the values of each of the following:
11. –1 –1 1
tan 2 cos 2sin
2
12. cot (tan–1a + cot–1 a)
13.
2
–1 –1
2 2
1 2 1
tan sin cos
2 1 1
x y
x y
−
+
+ +
, | x| < 1, y > 0 and xy < 1
14. If sin
–1 –11
sin cos 1
5
x
+ =
, then find the value of x
15. If
–1 –11 1
tan tan
2 2 4
x x
x x
− + π
+ =
− +
, then find the value of x
Find the values of each of the expressions in Exercises 16 to 18.
16.
–1 2
sin sin
3
π
17. –1 3
tan tan
4
π
18.
–1 –13 3
tan sin cot
5 2
+
19.
1 7
cos cos is equal to
6
− π
(A)
7
6
π
(B)
5
6
π
(C)
3
π
(D)
6
π
20. 1 1
sin sin ( )
3 2
−π
− −
is equal to
(A)
1
2
(B)
1
3
(C)
1
4
(D) 1
21.
1 1
tan 3 cot ( 3)− −
− − is equal to
(A) π (B)
2
π
− (C) 0 (D) 2 3
17. INVERSE TRIGONOMETRIC FUNCTIONS 49
Miscellaneous Examples
Example 9 Find the value of
1 3
sin (sin )
5
− π
Solution We know that 1
sin (sin )x x−
= . Therefore,
1 3 3
sin (sin )
5 5
− π π
=
But
3
,
5 2 2
π π π
∉ −
, which is the principal branch of sin–1
x
However
3 3 2
sin ( ) sin( ) sin
5 5 5
π π π
= π − = and
2
,
5 2 2
π π π
∈ −
Therefore
1 13 2 2
sin (sin ) sin (sin )
5 5 5
− −π π π
= =
Example 10 Show that
1 1 13 8 84
sin sin cos
5 17 85
− − −
− =
Solution Let 1 3
sin
5
x−
= and 1 8
sin
17
y−
=
Therefore
3
sin
5
x = and
8
sin
17
y =
Now 2 9 4
cos 1 sin 1
25 5
x x= − = − = (Why?)
and
2 64 15
cos 1 sin 1
289 17
y y= − = − =
We have cos (x–y) = cos x cos y + sin x sin y
=
4 15 3 8 84
5 17 5 17 85
× + × =
Therefore
1 84
cos
85
x y −
− =
Hence 1 1 13 8 84
sin sin cos
5 17 85
− − −
− =
18. 50 MATHEMATICS
Example 11 Show that 1 1 112 4 63
sin cos tan
13 5 16
− − −
+ + = π
Solution Let 1 1 112 4 63
sin , cos , tan
13 5 16
x y z− − −
= = =
Then
12 4 63
sin , cos , tan
13 5 16
x y z= = =
Therefore
5 3 12 3
cos , sin , tan and tan
13 5 5 4
x y x y= = = =
We have
tan tan
tan( )
1 tan tan
x y
x y
x y
+
+ =
−
12 3
635 4
12 3 161
5 4
+
= = −
− ×
Hence tan( ) tanx y z+ = −
i.e., tan (x + y) = tan (–z) or tan (x + y) = tan (π – z)
Therefore x + y = – z or x + y = π – z
Since x, y and z are positive, x + y ≠ – z (Why?)
Hence x + y + z = π or
–1 –1 –112 4 63
sin cos tan
13 5 16
+ + = π
Example 12 Simplify
–1 cos sin
tan
cos sin
a x b x
b x a x
−
+
, if
a
b
tan x > –1
Solution We have,
–1 cos sin
tan
cos sin
a x b x
b x a x
−
+
=
–1
cos sin
costan
cos sin
cos
a x b x
b x
b x a x
b x
−
+
=
–1
tan
tan
1 tan
a
x
b
a
x
b
−
+
=
–1 –1
tan tan (tan )
a
x
b
− =
–1
tan
a
x
b
−
19. INVERSE TRIGONOMETRIC FUNCTIONS 51
Example 13 Solve tan–1
2x + tan–1
3x =
4
π
Solution We have tan–1 2x + tan–1 3x =
4
π
or
–1 2 3
tan
1 2 3
x x
x x
+
− ×
=
4
π
i.e.
–1
2
5
tan
1 6
x
x
−
=
4
π
Therefore 2
5
1 6
x
x−
= tan 1
4
π
=
or 6x2
+ 5x – 1 = 0 i.e., (6x – 1) (x + 1) = 0
which gives x =
1
6
or x = – 1.
Since x = – 1 does not satisfy the equation, as the L.H.S. of the equation becomes
negative,
1
6
x = is the only solution of the given equation.
Miscellaneous Exercise on Chapter 2
Find the value of the following:
1.
–1 13
cos cos
6
π
2.
–1 7
tan tan
6
π
Prove that
3.
–1 –13 24
2sin tan
5 7
= 4.
–1 –1 –18 3 77
sin sin tan
17 5 36
+ =
5.
–1 –1 –14 12 33
cos cos cos
5 13 65
+ = 6.
–1 –1 –112 3 56
cos sin sin
13 5 65
+ =
7.
–1 –1 –163 5 3
tan sin cos
16 13 5
= +
8.
–1 1 1 11 1 1 1
tan tan tan tan
5 7 3 8 4
− − − π
+ + + =
20. 52 MATHEMATICS
Prove that
9.
–1 –11 1
tan cos
2 1
x
x
x
−
= +
, x ∈ [0, 1]
10. –1 1 sin 1 sin
cot
21 sin 1 sin
x x x
x x
+ + −
= + − −
, 0,
4
x
π
∈
11. –1 –11 1 1
tan cos
4 21 1
x x
x
x x
+ − − π
= − + + −
,
1
1
2
x− ≤ ≤ [Hint: Put x = cos 2θ]
12. 1 19 9 1 9 2 2
sin sin
8 4 3 4 3
− −π
− =
Solve the following equations:
13. 2tan–1
(cos x) = tan–1
(2 cosec x) 14.
–1 –11 1
tan tan ,( 0)
1 2
x
x x
x
−
= >
+
15. sin (tan–1
x), | x| < 1 is equal to
(A) 2
1
x
x−
(B) 2
1
1 x−
(C) 2
1
1 x+
(D) 2
1
x
x+
16. sin–1 (1 – x) – 2 sin–1 x =
2
π
, then x is equal to
(A) 0,
1
2
(B) 1,
1
2
(C) 0 (D)
1
2
17.
1 1
tan tan
x x y
y x y
− − −
− +
is equal to
(A)
2
π
(B)
3
π
(C)
4
π
(D)
3
4
− π
21. INVERSE TRIGONOMETRIC FUNCTIONS 53
Summary
The domains and ranges (principal value branches) of inverse trigonometric
functions are given in the following table:
Functions Domain Range
(Principal Value Branches)
y = sin–1
x [–1, 1] ,
2 2
−π π
y = cos–1
x [–1, 1] [0, π]
y = cosec–1
x R – (–1,1) ,
2 2
−π π
– {0}
y = sec–1 x R – (–1, 1) [0, π] – { }
2
π
y = tan–1
x R ,
2 2
π π
−
y = cot–1
x R (0, π)
sin–1
x should not be confused with (sin x)–1
. In fact (sin x)–1
=
1
sin x
and
similarly for other trigonometric functions.
The value of an inverse trigonometric functions which lies in its principal
value branch is called the principal value of that inverse trigonometric
functions.
For suitable values of domain, we have
y = sin–1 x ⇒ x = sin y x = sin y ⇒ y = sin–1 x
sin (sin–1 x) = x sin–1 (sin x) = x
sin–1
1
x
= cosec–1 x cos–1 (–x) = π – cos–1 x
cos–1
1
x
= sec–1x cot–1 (–x) = π – cot–1 x
tan–1
1
x
= cot–1 x sec–1 (–x) = π – sec–1 x
22. 54 MATHEMATICS
sin–1
(–x) = – sin–1
x tan–1
(–x) = – tan–1
x
tan–1
x + cot–1
x =
2
π
cosec–1
(–x) = – cosec–1
x
sin–1 x + cos–1 x =
2
π
cosec–1 x + sec–1 x =
2
π
tan–1
x + tan–1
y = tan–1
1
x y
xy
+
−
2tan–1
x = tan–1
2
21
x
x−
tan–1
x – tan–1
y = tan–1
1
x y
xy
−
+
2tan–1
x = sin–1
2
2
1
x
x+
= cos–1
2
2
1
1
x
x
−
+
Historical Note
The study of trigonometry was first started in India. The ancient Indian
Mathematicians, Aryabhatta (476A.D.), Brahmagupta (598 A.D.), Bhaskara I
(600A.D.) and Bhaskara II (1114A.D.) got important results of trigonometry.All
this knowledge went from India to Arabia and then from there to Europe. The
Greeks had also started the study of trigonometry but their approach was so
clumsy that when the Indian approach became known, it was immediately adopted
throughout the world.
In India, the predecessor of the modern trigonometric functions, known as
the sine of an angle, and the introduction of the sine function represents one of
the main contribution of the siddhantas (Sanskrit astronomical works) to
mathematics.
Bhaskara I (about 600A.D.) gave formulae to find the values of sine functions
for angles more than 90°. A sixteenth century Malayalam work Yuktibhasa
contains a proof for the expansion of sin (A + B). Exact expression for sines or
cosines of 18°, 36°, 54°, 72°, etc., were given by Bhaskara II.
The symbols sin–1
x, cos–1
x, etc., for arc sin x, arc cos x, etc., were suggested
by the astronomer Sir John F.W. Hersehel (1813) The name of Thales
(about 600 B.C.) is invariably associated with height and distance problems. He
is credited with the determination of the height of a great pyramid in Egypt by
measuring shadows of the pyramid and an auxiliary staff (or gnomon) of known
23. INVERSE TRIGONOMETRIC FUNCTIONS 55
height, and comparing the ratios:
H
S
h
s
= = tan (sun’s altitude)
Thales is also said to have calculated the distance of a ship at sea through
the proportionality of sides of similar triangles. Problems on height and distance
using the similarity property are also found in ancient Indian works.
— —