Inverse Trigonometric Functions Notes

Chapter 1 discussed the existence of the inverse of a function f, denoted as f^{-1}, which exists if f 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.

Review of trigonometric functions from Class XI:
- Sine function: sin : R \rightarrow [-1, 1]
- Cosine function: cos : R \rightarrow [-1, 1]
- Tangent function: tan : R - { x : x = (2n + 1) \frac{\pi}{2}, n \in Z } \rightarrow R
- Cotangent function: cot : R - { x : x = n\pi, n \in Z } \rightarrow R
- Secant function: sec : R - { x : x = (2n + 1) \frac{\pi}{2}, n \in Z } \rightarrow R - (-1, 1)
- Cosecant function: cosec : R - { x : x = n\pi, n \in Z } \rightarrow R - (-1, 1)
If f : X \rightarrow Y with f(x) = y is one-to-one and onto, a unique function g : Y \rightarrow X exists such that g(y) = x, where x \in X and y \in Y.
The domain of g equals the range of f, and the range of g equals the domain of f.
The function g is the inverse of f denoted by f^{-1}.
g is also one-to-one and onto, with the inverse of g being f. Thus, 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

The sine function has a domain of all real numbers and a range of [-1, 1].
Restricting the domain to [-\frac{\pi}{2}, \frac{\pi}{2}] makes it one-to-one and onto with a range of [-1, 1].
Sine function restricted to intervals such as [-\frac{3\pi}{2}, -\frac{\pi}{2}], [-\frac{\pi}{2}, \frac{\pi}{2}],[\frac{\pi}{2}, \frac{3\pi}{2}] etc., is one-to-one with range [-1, 1].
The inverse of the sine function is denoted as sin^{-1} (arc sine function).
sin^{-1} has a domain of [-1, 1] and a range in any of the above intervals.
Each interval corresponds to a branch of the sin^{-1} function.
The branch with range [-\frac{\pi}{2}, \frac{\pi}{2}] is the principal value branch.
When referring to sin^{-1}, it is generally taken as the function with domain [-1, 1] and range [-\frac{\pi}{2}, \frac{\pi}{2}].
sin^{-1} : [-1, 1] \rightarrow [-\frac{\pi}{2}, \frac{\pi}{2}]
From the definition of inverse functions:
- sin(sin^{-1} x) = x if -1 \le x \le 1
- sin^{-1}(sin x) = x if \frac{-\pi}{2} \le x \le \frac{\pi}{2}

Remarks

If y = f(x) is invertible, then x = f^{-1}(y).
The graph of sin^{-1} can be derived from sin x by interchanging the x and y axes.
If (a, b) is on the graph of sin x, then (b, a) is on the graph of sin^{-1} x.
The graph of an inverse function is a mirror image of the original function along the line y = x.

Cosine function has a domain of all real numbers and a range of [-1, 1].
Restricting the domain to [0, \pi] makes it one-to-one and onto with a range of [-1, 1].
Cosine function restricted to any of the intervals [-\pi, 0], [0, \pi], [\pi, 2\pi] etc., is bijective with range [-1, 1].
The inverse of the cosine function is denoted by cos^{-1} (arc cosine function).
cos^{-1} has a domain of [-1, 1] and range could be any of the intervals [-\pi, 0], [0, \pi], [\pi, 2\pi] etc.
The branch with range [0, \pi] is the principal value branch of the function cos^{-1}.
cos^{-1} : [-1, 1] \rightarrow [0, \pi].
The graph of y = cos^{-1} x can be drawn similarly to y = sin^{-1} x.

Inverse Cosecant Function

Since cosec x = \frac{1}{sin x}, the domain is {x : x \in R \text{ and } x \neq n\pi, n \in Z }, and the range is {y : y \in R, y \geq 1 \text{ or } y \leq -1}, which is R - (-1, 1).
y = cosec x assumes all real values except -1 < y < 1 and is undefined for integral multiples of \pi.
Restricting the domain to [-\frac{\pi}{2}, \frac{\pi}{2}] - {0} makes it one-to-one and onto with range R - (-1, 1).
Cosec function restricted to any of the intervals [-\frac{3\pi}{2}, -\frac{-\pi}{2}] - {-\pi}, [-\frac{\pi}{2}, \frac{\pi}{2}] - {0}, [\frac{3\pi}{2}, \frac{\pi}{2}] - {\pi} etc., is bijective and its range is the set of all real numbers R - (-1, 1).
cosec^{-1} can be defined as a function with domain R - (-1, 1) and range in any of the intervals [-\frac{3\pi}{2}, -\frac{-\pi}{2}] - {-\pi}, [-\frac{\pi}{2}, \frac{\pi}{2}] - {0}, [\frac{3\pi}{2}, \frac{\pi}{2}] - {\pi} etc.
The function corresponding to the range [-\frac{\pi}{2}, \frac{\pi}{2}] - {0} is the principal value branch of cosec^{-1}.
cosec^{-1} : R - (-1, 1) \rightarrow [-\frac{\pi}{2}, \frac{\pi}{2}] - {0}

Inverse Secant Function

Since sec x = \frac{1}{cos x}, the domain of y = sec x is R - {x : x = (2n + 1) \frac{\pi}{2}, n \in Z } and range is R - (-1, 1).
sec x assumes all real values except -1 < y < 1 and is not defined for odd multiples of \frac{\pi}{2}.
Restricting the domain to [0, \pi] - {\frac{\pi}{2}} makes it one-to-one and onto with its range as the set R - (-1, 1).
Secant function restricted to any of the intervals [-\pi, 0] - {-\frac{\pi}{2}}, [0, \pi] - {\frac{\pi}{2}}, [\pi, 2\pi] - {\frac{3\pi}{2}} etc., is bijective and its range is R - {-1, 1}.
sec^{-1} can be defined as a function with domain R - (-1, 1) and range could be any of the intervals [-\pi, 0] - {-\frac{\pi}{2}}, [0, \pi] - {\frac{\pi}{2}}, [\pi, 2\pi] - {\frac{3\pi}{2}} etc.
The branch with range [0, \pi] - {\frac{\pi}{2}} is called the principal value branch of the function sec^{-1}.
sec^{-1} : R - (-1,1) \rightarrow [0, \pi] - {\frac{\pi}{2}}

Inverse Tangent Function

The domain of the tangent function is {x : x \in R \text{ and } x \neq (2n +1) \frac{\pi}{2}, n \in Z } and the range is R.
The tangent function is not defined for odd multiples of \frac{\pi}{2}.
Restricting the domain to (-\frac{\pi}{2}, \frac{\pi}{2}) makes it one-to-one and onto with its range as R.
Tangent function restricted to any of the intervals (-\frac{3\pi}{2}, -\frac{\pi}{2}), (-\frac{\pi}{2}, \frac{\pi}{2}), (\frac{\pi}{2}, \frac{3\pi}{2}) etc., is bijective and its range is R.
tan^{-1} can be defined as a function whose domain is R and range could be any of the intervals (-\frac{3\pi}{2}, -\frac{\pi}{2}), (-\frac{\pi}{2}, \frac{\pi}{2}), (\frac{\pi}{2}, \frac{3\pi}{2}), and so on.
The branch with range (-\frac{\pi}{2}, \frac{\pi}{2}) is the principal value branch of the function tan^{-1}.
tan^{-1} : R \rightarrow (-\frac{\pi}{2}, \frac{\pi}{2})

Inverse Cotangent Function

The domain of the cotangent function is {x : x \in R \text{ and } x \neq n\pi, n \in Z } and range is R.
The cotangent function is not defined for integral multiples of \pi.
Restricting the domain to (0, \pi) makes it bijective with its range as R.
Cotangent function restricted to any of the intervals (-\pi, 0), (0, \pi), (\pi, 2\pi) etc., is bijective and its range is R.
cot^{-1} can be defined as a function whose domain is the R and range as any of the intervals (-\pi, 0), (0, \pi), (\pi, 2\pi) etc.
The function with range (0, \pi) is called the principal value branch of the function cot^{-1}.
cot^{-1} : R \rightarrow (0, \pi)

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

Note

sin^{-1}x should not be confused with (sin x)^{-1}. In fact (sin x)^{-1} = \frac{1}{sin x} and similarly for other trigonometric functions.
Whenever no branch of an inverse trigonometric functions is mentioned, we mean the principal value branch of that function.
The value of an inverse trigonometric functions which lies in