Inverse Trigonometric Functions Notes

Mathematics and Inverse Trigonometric Functions
Introduction

  • Chapter 1 discussed the existence of the inverse of a function ff, denoted as f−1f^{-1}, which exists if ff is one-to-one and onto.

  • Trigonometric functions were identified as not being one-to-one and onto over their natural domains and ranges in Class XI, implying their inverses do not exist under those conditions.

  • This chapter focuses on domain and range restrictions for trigonometric functions to ensure the existence of their inverses.

  • It includes graphical representations and elementary properties.

  • Inverse trigonometric functions are crucial in calculus for defining integrals and are applied in science and engineering.

Basic Concepts

  • Review of trigonometric functions from Class XI:

    • Sine function: sin:R→[−1,1]sin : R \rightarrow [-1, 1]

    • Cosine function: cos:R→[−1,1]cos : R \rightarrow [-1, 1]

    • Tangent function: tan:R−x:x=(2n+1)π2,n∈Z→Rtan : R - { x : x = (2n + 1) \frac{\pi}{2}, n \in Z } \rightarrow R

    • Cotangent function: cot:R−x:x=nπ,n∈Z→Rcot : R - { x : x = n\pi, n \in Z } \rightarrow R

    • Secant function: sec:R−x:x=(2n+1)π2,n∈Z→R−(−1,1)sec : R - { x : x = (2n + 1) \frac{\pi}{2}, n \in Z } \rightarrow R - (-1, 1)

    • Cosecant function: cosec:R−x:x=nπ,n∈Z→R−(−1,1)cosec : R - { x : x = n\pi, n \in Z } \rightarrow R - (-1, 1)

  • If f:X→Yf : X \rightarrow Y with f(x)=yf(x) = y is one-to-one and onto, a unique function g:Y→Xg : Y \rightarrow X exists such that g(y)=xg(y) = x, where x∈Xx \in X and y∈Yy \in Y.

  • The domain of gg equals the range of ff, and the range of gg equals the domain of ff.

  • The function gg is the inverse of ff denoted by f−1f^{-1}.

  • gg is also one-to-one and onto, with the inverse of gg being ff. Thus, g−1=(f−1)−1=fg^{-1} = (f^{-1})^{-1} = f.

    • (f−1of)(x)=f−1(f(x))=f−1(y)=x(f^{-1} o f)(x) = f^{-1}(f(x)) = f^{-1}(y) = x

    • (fof−1)(y)=f(f−1(y))=f(x)=y(f o f^{-1})(y) = f(f^{-1}(y)) = f(x) = y

Inverse Sine Function

  • The sine function has a domain of all real numbers and a range of [−1,1][-1, 1].

  • Restricting the domain to [−π2,π2][-\frac{\pi}{2}, \frac{\pi}{2}] makes it one-to-one and onto with a range of [−1,1][-1, 1].

  • Sine function restricted to intervals such as [−3π2,−π2][-\frac{3\pi}{2}, -\frac{\pi}{2}], [−π2,π2][-\frac{\pi}{2}, \frac{\pi}{2}],[π2,3π2][\frac{\pi}{2}, \frac{3\pi}{2}] etc., is one-to-one with range [−1,1][-1, 1].

  • The inverse of the sine function is denoted as sin−1sin^{-1} (arc sine function).

  • sin−1sin^{-1} has a domain of [−1,1][-1, 1] and a range in any of the above intervals.

  • Each interval corresponds to a branch of the sin−1sin^{-1} function.

  • The branch with range [−π2,π2][-\frac{\pi}{2}, \frac{\pi}{2}] is the principal value branch.

  • When referring to sin−1sin^{-1}, it is generally taken as the function with domain [−1,1][-1, 1] and range [−π2,π2][-\frac{\pi}{2}, \frac{\pi}{2}].

  • sin−1:[−1,1]→[−π2,π2]sin^{-1} : [-1, 1] \rightarrow [-\frac{\pi}{2}, \frac{\pi}{2}]

  • From the definition of inverse functions:

    • sin(sin−1x)=xsin(sin^{-1} x) = x if −1≤x≤1-1 \le x \le 1

    • sin−1(sinx)=xsin^{-1}(sin x) = x if −π2≤x≤π2\frac{-\pi}{2} \le x \le \frac{\pi}{2}

Remarks

  • If y=f(x)y = f(x) is invertible, then x=f−1(y)x = f^{-1}(y).

  • The graph of sin−1sin^{-1} can be derived from sinxsin x by interchanging the x and y axes.

  • If (a,b)(a, b) is on the graph of sinxsin x, then (b,a)(b, a) is on the graph of sin−1xsin^{-1} x.

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

Inverse Cosine Function

  • Cosine function has a domain of all real numbers and a range of [−1,1][-1, 1].

  • Restricting the domain to [0,π][0, \pi] makes it one-to-one and onto with a range of [−1,1][-1, 1].

  • Cosine function restricted to any of the intervals [−π,0][-\pi, 0], [0,π][0, \pi], [π,2π][\pi, 2\pi] etc., is bijective with range [−1,1][-1, 1].

  • The inverse of the cosine function is denoted by cos−1cos^{-1} (arc cosine function).

  • cos−1cos^{-1} has a domain of [−1,1][-1, 1] and range could be any of the intervals [−π,0][-\pi, 0], [0,π][0, \pi], [π,2π][\pi, 2\pi] etc.

  • The branch with range [0,π][0, \pi] is the principal value branch of the function cos−1cos^{-1}.

  • cos−1:[−1,1]→[0,π]cos^{-1} : [-1, 1] \rightarrow [0, \pi].

  • The graph of y=cos−1xy = cos^{-1} x can be drawn similarly to y=sin−1xy = sin^{-1} x.

Inverse Cosecant and Secant Functions

Inverse Cosecant Function

  • Since cosecx=1sinxcosec x = \frac{1}{sin x}, the domain is x:x∈R and x≠nπ,n∈Z{x : x \in R \text{ and } x \neq n\pi, n \in Z }, and the range is y:y∈R,y≥1 or y≤−1{y : y \in R, y \geq 1 \text{ or } y \leq -1}, which is R−(−1,1)R - (-1, 1).

  • y=cosecxy = cosec x assumes all real values except -1 < y < 1 and is undefined for integral multiples of π\pi.

  • Restricting the domain to [−π2,π2]−0[-\frac{\pi}{2}, \frac{\pi}{2}] - {0} makes it one-to-one and onto with range R−(−1,1)R - (-1, 1).

  • Cosec function restricted to any of the intervals [−3π2,−−π2]−−π[-\frac{3\pi}{2}, -\frac{-\pi}{2}] - {-\pi}, [−π2,π2]−0[-\frac{\pi}{2}, \frac{\pi}{2}] - {0}, [3π2,π2]−π[\frac{3\pi}{2}, \frac{\pi}{2}] - {\pi} etc., is bijective and its range is the set of all real numbers R−(−1,1)R - (-1, 1).

  • cosec−1cosec^{-1} can be defined as a function with domain R−(−1,1)R - (-1, 1) and range in any of the intervals [−3π2,−−π2]−−π[-\frac{3\pi}{2}, -\frac{-\pi}{2}] - {-\pi}, [−π2,π2]−0[-\frac{\pi}{2}, \frac{\pi}{2}] - {0}, [3π2,π2]−π[\frac{3\pi}{2}, \frac{\pi}{2}] - {\pi} etc.

  • The function corresponding to the range [−π2,π2]−0[-\frac{\pi}{2}, \frac{\pi}{2}] - {0} is the principal value branch of cosec−1cosec^{-1}.

  • cosec−1:R−(−1,1)→[−π2,π2]−0cosec^{-1} : R - (-1, 1) \rightarrow [-\frac{\pi}{2}, \frac{\pi}{2}] - {0}

Inverse Secant Function

  • Since secx=1cosxsec x = \frac{1}{cos x}, the domain of y=secxy = sec x is R−x:x=(2n+1)π2,n∈ZR - {x : x = (2n + 1) \frac{\pi}{2}, n \in Z } and range is R−(−1,1)R - (-1, 1).

  • secxsec x assumes all real values except -1 < y < 1 and is not defined for odd multiples of π2\frac{\pi}{2}.

  • Restricting the domain to [0,π]−π2[0, \pi] - {\frac{\pi}{2}} makes it one-to-one and onto with its range as the set R−(−1,1)R - (-1, 1).

  • Secant function restricted to any of the intervals [−π,0]−−π2[-\pi, 0] - {-\frac{\pi}{2}}, [0,π]−π2[0, \pi] - {\frac{\pi}{2}}, [π,2π]−3π2[\pi, 2\pi] - {\frac{3\pi}{2}} etc., is bijective and its range is R−−1,1R - {-1, 1}.

  • sec−1sec^{-1} can be defined as a function with domain R−(−1,1)R - (-1, 1) and range could be any of the intervals [−π,0]−−π2[-\pi, 0] - {-\frac{\pi}{2}}, [0,π]−π2[0, \pi] - {\frac{\pi}{2}}, [π,2π]−3π2[\pi, 2\pi] - {\frac{3\pi}{2}} etc.

  • The branch with range [0,π]−π2[0, \pi] - {\frac{\pi}{2}} is called the principal value branch of the function sec−1sec^{-1}.

  • sec−1:R−(−1,1)→[0,π]−π2sec^{-1} : R - (-1,1) \rightarrow [0, \pi] - {\frac{\pi}{2}}

Inverse Tangent and Cotangent Functions

Inverse Tangent Function

  • The domain of the tangent function is x:x∈R and x≠(2n+1)π2,n∈Z{x : x \in R \text{ and } x \neq (2n +1) \frac{\pi}{2}, n \in Z } and the range is RR.

  • The tangent function is not defined for odd multiples of π2\frac{\pi}{2}.

  • Restricting the domain to (−π2,π2)(-\frac{\pi}{2}, \frac{\pi}{2}) makes it one-to-one and onto with its range as RR.

  • Tangent function restricted to any of the intervals (−3π2,−π2)(-\frac{3\pi}{2}, -\frac{\pi}{2}), (−π2,π2)(-\frac{\pi}{2}, \frac{\pi}{2}), (π2,3π2)(\frac{\pi}{2}, \frac{3\pi}{2}) etc., is bijective and its range is RR.

  • tan−1tan^{-1} can be defined as a function whose domain is RR and range could be any of the intervals (−3π2,−π2)(-\frac{3\pi}{2}, -\frac{\pi}{2}), (−π2,π2)(-\frac{\pi}{2}, \frac{\pi}{2}), (π2,3π2)(\frac{\pi}{2}, \frac{3\pi}{2}), and so on.

  • The branch with range (−π2,π2)(-\frac{\pi}{2}, \frac{\pi}{2}) is the principal value branch of the function tan−1tan^{-1}.

  • tan−1:R→(−π2,π2)tan^{-1} : R \rightarrow (-\frac{\pi}{2}, \frac{\pi}{2})

Inverse Cotangent Function

  • The domain of the cotangent function is x:x∈R and x≠nπ,n∈Z{x : x \in R \text{ and } x \neq n\pi, n \in Z } and range is RR.

  • The cotangent function is not defined for integral multiples of π\pi.

  • Restricting the domain to (0,π)(0, \pi) makes it bijective with its range as RR.

  • Cotangent function restricted to any of the intervals (−π,0)(-\pi, 0), (0,π)(0, \pi), (π,2π)(\pi, 2\pi) etc., is bijective and its range is RR.

  • cot−1cot^{-1} can be defined as a function whose domain is the RR and range as any of the intervals (−π,0)(-\pi, 0), (0,π)(0, \pi), (π,2π)(\pi, 2\pi) etc.

  • The function with range (0,π)(0, \pi) is called the principal value branch of the function cot−1cot^{-1}.

  • cot−1:R→(0,π)cot^{-1} : R \rightarrow (0, \pi)

Table of Inverse Trigonometric Functions

The following table summarizes the inverse trigonometric functions, including their domains and ranges (principal value branches):

Note

  1. sin−1xsin^{-1}x should not be confused with (sinx)−1(sin x)^{-1}. In fact (sinx)−1=1sinx(sin x)^{-1} = \frac{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