Chapter 2 Inverse Trigonometric functions

• Inverse trigonometric functions are covered in Chapter 2.
• Mathematics is fundamentally the science of self-evident things.

Introduction

• Inverses exist for one-to-one and onto functions (Chapter 1).
• Trigonometric functions are not one-to-one and onto in natural domains.
• The chapter focuses on restricting trigonometric function domains and ranges to allow inverses, graphical representations and elementary properties.
• Inverse trigonometric functions have a vital role in calculus because they define many integrals and are used in science and engineering.

Basic Concepts

• 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)

• If f : X→Y is one-one and onto such that f(x) = y, then there is a unique function g : Y→X where g(y) = x for x ∈ X, y ∈ Y.

• The domain of g = range of f and the range of g = domain of f.

• The function g is the inverse of f, denoted by f –1 and g is one-to-one and onto and the inverse of g is f.

• g –1 = (f –1)–1 = f

• (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

• Restricting the sine function's domain to [-π/2, π/2] makes it one-to-one and onto with a range of [-1, 1].

• The sine function is one-to-one with a range of [-1, 1] when limited to any of the intervals [-3π/2, -π/2], [-π/2, π/2], [π/2, 3π/2], etc.

• The inverse of the sine function (arcsine), denoted by sin-1, is then defined.

• sin-1 domain is [-1, 1]

• range can be any of the intervals such as [-3π/2, -π/2], [-π/2, π/2], [π/2, 3π/2]

• Each interval corresponds to a branch of the sin-1 function.

• The principal value branch has a range of [-π/2, π/2].

• Unless otherwise specified, sin-1 refers to the function with domain [-1, 1] and range [-π/2, π/2].

• sin(sin-1x) = x if -1 ≤ x ≤ 1

• sin-1(sin x) = x if -π/2 ≤ x ≤ π/2.

• If y = sin-1x, sin y = x.

• The graph of sin-1 function is obtained by interchanging x and y axes from the graph of the original function.

• If (a, b) is a point on the graph of sine function, then (b, a) becomes the corresponding point on the graph of the inverse.

• Graphs of y = sin x and y = sin-1 x can be obtained by interchanging x and y axes.

• The dark portion of the graph of y = sin-1 x shows the principal value branch.

• A graph of an inverse function is a mirror image of the original function along the line y = x.

• Cosine Function

  • Similar to the sine function, the cosine function has a domain of all real numbers and a range of [-1, 1].
  • Restricting the cosine function's domain to [0, π] makes it one-to-one and onto with a range of [-1, 1].
  • The inverse of the cosine function is denoted by cos-1 (arc cosine function).
  • The domain of cos-1 is [-1, 1]
  • Range could be any of the intervals [-π, 0], [0, π], [π, 2π] etc.
  • The branch with range [0, π] is the principal value branch.
  • cos-1 : [-1, 1] → [0, π].

• The graphs of y = cos x and y = cos-1 x can be obtained by following the method used for sine function.

• Since cosec x = 1/sin x, the domain of the cosec function is {x: x ∈ R and x ≠ nπ, n ∈ Z} and the range is {y : y ∈ R, y ≥ 1 or y ≤ –1} i.e., the set R – (–1, 1).

  • cosec x assumes values except –1 < y < 1
  • It is undefined for integral multiple of π.

• The cosec function is one to one and onto with range R – (-1, 1) if we domain is restricted to [−π/2, π/2] – {0}.

• Cosecant function restricted to any of the intervals [-3π/2, -π/2] − {−π},[−π/2, π/2] − {0}, [π/2, 3π/2] − {π} etc., is bijective and its range is the set of all real numbers R – (-1, 1).

• The domain of cosec-1 can be defined as R – (-1, 1) where the range could be any of the intervals [−3π/2, -π/2] − {−π},[−π/2, π/2] − {0}, [π/2, 3π/2] − {π} etc.

• The range corresponding to [−π/ 2, π/22] – {0} is the principal value branch

• cosec-1 : R – (–1, 1) → [−π/2, π/2] – {0}

• Secant Function

• sec x = 1/cos x

• The domain of y = sec x is the set R – {x : x = (2n + 1)(π/2), n ∈ Z}

• The range is the set R – (–1, 1)

• sec (secant function) assumes all real values except –1 < y < 1 and is not defined for odd multiples of π/2.

• Restricting the domain of secant function to [0, π] – {π/2}, makes it one-to-one and onto with its range R – (–1, 1).

• Secant is restricted to any of the intervals [–π, 0] – {−π/2}, [0, π] – {π/2} , [π, 2π] – {3π/2} etc., is bijective and its range is R – {–1, 1}.

• sec-1 can be defined as a function whose domain is R – (–1, 1)

• Range can be any of the intervals [–π, 0] – {−π/2}, [0, π] – {π/2} , [π, 2π] – {3π/2} etc.

• Principal value branch has a range [0, π] – {π/2}

  • sec-1 : R – (–1,1) → [0, π] – {π/2}

• Tangent Function

  • The domain of the tangent function is {x : x ∈ R and x ≠ (2n +1)(π/2), n ∈ Z}
  • The range is R
  • tan function is not defined for odd multiples of π/2

• Restricting the tangent function domain to (−π/2, π/2), then it is one-one and onto with its range as R.

• Tangent function restricted to any of the intervals (−3π/2, −π/2), (−π/2, π/2), (π/2, 3π/2) etc., is bijective and its range is R.

• tan-1 can be defined as a function whose domain is R where the range could be any of the intervals (−3π/2, −π/2), (−π/2, π/2), (π/2, 3π/2) and so on.

  • Principal value branch has range (−π/2, π/2)
  • tan-1 : R → (−π/2, π/2)

• Cotangent Function

  • The domain of the cotangent function is {x : x ∈ R and x ≠ nπ, n ∈ Z} and range is R
  • cotangent function is not defined for integral multiples of π
  • Limiting the domain of cotangent function to (0, π) makes it bijective with a range of R.
  • cotangent function restricted to any of the intervals (–π, 0), (0, π), (π, 2π) etc., is bijective and its range is R.
  • cot -1 can be defined as a function whose domain is R where the range as any of the intervals
  • The function with range (0, π) is called the principal value branch
  • cot-1 : R → (0, π)

• sin-1x 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

  • 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 note

• Trigonometry study started in India.
• Ancient Indian Mathematicians like Aryabhata, Brahmagupta, Bhaskara I and Bhaskara II, made important discoveries in trigonometry and then the knowledge spread.
• The Greeks also studied trigonometry but when the Indian way was known, it was quickly used worldwide.
• The sine of an angle came from India and is one of the main things from the siddhantas (Sanskrit astronomical works) for math.
• Bhaskara I gave ways to find sine values for angles bigger than 90°.
• A book from the 16th century in Malayalam language called Yuktibhasa, explains the expansion of sin (A + B).
• Bhaskara II gave exact statements for sines or cosines of 18°, 36°, 54°, 72°, etc.
• Sir John F.W. Hersehen suggested the symbols sin–1 x and cos–1x,in 1813 to be used for arc sin x and arc cos x.
• Thales (around 600 B.C.) is known for height and distance problems.
• He measured a pyramid's height in Egypt by using ratios of shadows and staff
• (H/S) = (h/s) = tan (sun’s altitude)
• Thales calculated ship distance at sea using triangle sides.
• Height and distance problems are also in ancient Indian works.