Inverse Trigonometric Functions: Chapter 2

• Chapter 2 focuses on inverse trigonometric functions
• Mathematics, in general, is the science of self-evident things, according to Felix Klein

Introduction

• Inverse of function f, denoted as f^-1, exists if f is one-to-one and onto
• Trigonometric functions are not one-to-one and onto over their natural domains and ranges
• Chapter explores restrictions on domains/ranges of trigonometric functions

Inverse Trigonometric Functions

• Important in calculus because they define many integrals
• Used in science and engineering

Basic Concepts of Trigonometric Functions

• Sine function: sine : R -> [-1, 1]
• Cosine function: cos : R -> [-1, 1]
• Tangent function: tan : R – { x : x = (2n + 1)(π/2), n ∈ Z} → R
• Cotangent function: cot : R – { x : x = nπ, n ∈ Z} → R
• Secant function: sec : R – { x : x = (2n + 1)(π/2), n ∈ Z} → R – (– 1, 1)
• Cosecant function: cosec : R – { x : x = nπ, n ∈ Z} → R – (– 1, 1)

Inverse Function Basics

• If f: X→Y and f(x) = y is one-to-one and onto, there's a unique function g: Y→X where g(y) = x
• Domain of g equals the range of f, and range of g equals domain of f
• Function g is the inverse of f, denoted as f^-1; g is also one-to-one and onto
• g^-1 = (f^-1)^-1 = f
• (f^-1 o f)(x) = f^-1(f(x)) = f^-1(y) = x
• (f o f^-1)(y) = f(f^-1(y)) = f(x) = y

Sine Function Domain Restrictions

• Domain of sine function is all real numbers, range is the closed interval [-1, 1]
• Restricting domain to [-π/2, π/2] makes it one-to-one and onto with range [-1, 1]
• Sine function remains one-to-one over intervals: [-3π/2, -π/2], [-π/2, π/2], [π/2, 3π/2], etc., with range [-1, 1]

Inverse Sine Function

• Inverse of the sine function can be defined in each of these intervals.
• Denoted by sin^-1 (arcsin)
• sin^-1 is a function with domain [-1, 1] and range within intervals such as [-3π/2, -π/2], [-π/2, π/2], or [π/2, 3π/2]
• Each interval corresponds to a branch of the sin^-1 function
• Principal value branch has a range of [-π/2, π/2]
• When referring to sin^-1, it is function with domain [-1, 1] and range [-π/2, π/2]
• sin^-1 : [-1, 1] -> [-π/2, π/2]
• sin(sin^-1(x)) = x if -1 ≤ x ≤ 1
• sin^-1(sin(x)) = x if -π/2 ≤ x ≤ π/2, which means if y = sin^-1(x), then sin y = x.

Graphing Inverse Sine Function

• Graph of sin^-1 function is obtained by interchanging the x and y axes of the original sine function
• If (a, b) is a point on the sine function graph, (b, a) becomes the corresponding point on the inverse's graph
• Graph of an inverse function mirrors (reflects) along the line y = x
• Visualize the graphs of y = sin x and y = sin^-1 x together on the same axes

Cosine Function

• Similar to sine, the cosine function has a domain of all real numbers and a range of [-1, 1]
• Restricting the domain to [0, π] makes the cosine function one-to-one and onto with a range of [-1, 1]
• Cosine function can be restricted to any of the intervals [-π, 0], [0, π], [π, 2π]

Inverse Cosine Function

• cos^-1 is a function with a domain of [-1, 1]
• Range can be any of the intervals [-π, 0], [0, π], [π, 2π]
• A branch of the function cos^-1 is gotten with each such interval
• A range of [0, π] is defined as the principal value branch of cos^-1.
• cos^-1 : [-1, 1] -> [0, π].
• The graph of y = cos^-1 x can be drawn the same way as the graph of y = sin^-1 x
• The graphs of y = cos x and y = cos^-1 x are given in Fig 2.2 (i) and (ii).

Cosecant and Secant Functions

• cosec x = 1 / sin x
• Cosecant: domain is {x : x ∈ R and x ≠ nπ, n ∈ Z} and the range is {y : y ∈ R, y ≥ 1 or y ≤ –1}, expressed as R – (–1, 1)
• This means cosec x assumes values except -1 < y < 1 and is undefined for integral multiples of π
• Restricting the cosec function to [−π/2, π/2] – {0} makes it one to one and onto with a range of R – (– 1, 1)
• Restricting the cosec function to any of the intervals [−3π/2, −π/2] – {−π}, [π/2, π/2] – {0}, [π/2, 3π/2] – {π}, is bijective and its range is the set of all real numbers R – (–1, 1).
• Cosec^-1 can be defined, the domain is R – (-1, 1) and the range could be any relevant interval
• Range corresponding to the interval [-π/2, π/2] – {0} is the principal value branch of cosec^-1
• cosec^-1 : R – (–1, 1) -> [-π/2, π/2] – {0}
• Secx, which implies that the domain of y = sec x is R – {x : x = (2n + 1)(π/2), n ∈ Z}, where n ∈ Z
• The range of secant is R – (–1, 1)
• This implies that sec (secant function) assumes all real values except –1 < y < 1 and is not defined for odd multiples of π/2
• You restrict the domain of secant function to [0, π] – {π/2}, then it is one-one and onto with

Tangent and Cotangent

• Domain of tangent is {x : x ∈ R and x ≠ (2n +1)(π/2), n ∈ Z}
• Range of tangent is R
• Tangent function is not defined for old multiples of π/2
• Restricting the domain of the function to (-π/2, π/2), then one-to- one and the range will be R.
• tan–1 : R → (-π/2, π/2)
• Domain of cotangent is {x : x ∈ R and x ≠ nπ, n ∈ Z}
• Range is R
• If domain is restricted to (0, π), it is bijective with a range of R
• cot–1 can be defined as a function whose domain is the R and range as any of the intervals
• cot–1 : R → (0, π)

Trigonometric Functions Table

• sin^-1: Domain [-1, 1], Range [-π/2, π/2]
• cos^-1: Domain [-1, 1], Range [0, π]
• cosec^-1: Domain R – (-1, 1), Range [-π/2, π/2] – {0}
• sec^-1: Domain R – (-1, 1), Range [0, π] – {π/2}
• tan^-1: Domain R, Range (-π/2, π/2)
• cot^-1: Domain R, Range (0, π)

Important Notes

• sin^-1(x) should not be confused with (sin x)^-1
• Whenever no branch of an inverse trigonometric functions is mentioned, mean the value is on the principal value branch
• The inverse trigonometric functions value is called the principal value of that inverse trigonometric functions when is resides in the range of the principal branch

Sine and Inverse Sine Relations

• y = sin^-1 x ⇒ x = sin y
• x = sin y ⇒ y = sin^-1 x
• sin (sin^-1 x) = x
• sin^-1 (sin x) = x

Historical Notes on Trigonometry

• Trigonometry study started in India
• Ancient Indian Mathematicians (Aryabhata, Brahmagupta, Bhaskara I & II) made important contributions
• Knowledge spread from India to Arabia, then to Europe
• Indians came up with the sine function and that become a main component of siddhantas
• Bhaskara I (600 A.D.) produced formulas for sine values of angles > 90 degrees
• Yuktibhasa work has a proof for sine expansion of sin (A + B); Bhaskara II discovered expressions for sines/cosines
• Astronomer Sir John F.W. Hersehel (1813) suggested the symbols sin–1 x, cos–1 x, for arc sin x, arc cos x
• Thales linked with height and distances, determined pyramid height using shadows and auxiliary staff
• Calculated ship distance using similar triangle proportionality, also found in ancient India