Inverse Trigonometric Functions - Properties, Domain, Range, Graphs - Free Printable

Problem Analysis:

The task involves understanding the domain and range of various inverse trigonometric functions. The table provided lists six inverse trigonometric functions along with their domains and ranges. The goal is to verify and explain the domain and range for each function.

Explanation of Each Function:

#### (i) \( y = \sin^{-1} x \)
- Domain: \([-1, 1]\)
- The sine function, \( \sin x \), has a range of \([-1, 1]\). Therefore, the input to the inverse sine function, \( \sin^{-1} x \), must lie within this interval.
- Range: \([- \frac{\pi}{2}, \frac{\pi}{2}]\)
- The inverse sine function, \( \sin^{-1} x \), returns an angle \( y \) such that \( \sin y = x \) and \( y \) lies in the interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\). This interval ensures that the function is one-to-one and thus invertible.

#### (ii) \( y = \cos^{-1} x \)
- Domain: \([-1, 1]\)
- Similar to the sine function, the cosine function, \( \cos x \), has a range of \([-1, 1]\). Therefore, the input to the inverse cosine function, \( \cos^{-1} x \), must also lie within this interval.
- Range: \([0, \pi]\)
- The inverse cosine function, \( \cos^{-1} x \), returns an angle \( y \) such that \( \cos y = x \) and \( y \) lies in the interval \([0, \pi]\). This interval ensures that the function is one-to-one and thus invertible.

#### (iii) \( y = \tan^{-1} x \)
- Domain: \(\mathbb{R}\)
- The tangent function, \( \tan x \), is defined for all real numbers except where \( \cos x = 0 \) (i.e., at \( x = \frac{\pi}{2} + k\pi \) for integers \( k \)). However, the range of \( \tan x \) is all real numbers, so the input to the inverse tangent function, \( \tan^{-1} x \), can be any real number.
- Range: \((- \frac{\pi}{2}, \frac{\pi}{2})\)
- The inverse tangent function, \( \tan^{-1} x \), returns an angle \( y \) such that \( \tan y = x \) and \( y \) lies in the interval \((- \frac{\pi}{2}, \frac{\pi}{2})\). This interval ensures that the function is one-to-one and thus invertible.

#### (iv) \( y = \cot^{-1} x \)
- Domain: \(\mathbb{R}\)
- The cotangent function, \( \cot x \), is defined for all real numbers except where \( \sin x = 0 \) (i.e., at \( x = k\pi \) for integers \( k \)). However, the range of \( \cot x \) is all real numbers, so the input to the inverse cotangent function, \( \cot^{-1} x \), can be any real number.
- Range: \((0, \pi)\)
- The inverse cotangent function, \( \cot^{-1} x \), returns an angle \( y \) such that \( \cot y = x \) and \( y \) lies in the interval \((0, \pi)\). This interval ensures that the function is one-to-one and thus invertible.

#### (v) \( y = \csc^{-1} x \)
- Domain: \((-\infty, -1] \cup [1, \infty)\)
- The cosecant function, \( \csc x \), is the reciprocal of the sine function, \( \sin x \). Since \( \sin x \) ranges from \([-1, 1]\), \( \csc x \) is defined for \( x \leq -1 \) or \( x \geq 1 \).
- Range: \([- \frac{\pi}{2}, \frac{\pi}{2}] \setminus \{0\}\)
- The inverse cosecant function, \( \csc^{-1} x \), returns an angle \( y \) such that \( \csc y = x \) and \( y \) lies in the interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\) excluding \( y = 0 \). This interval ensures that the function is one-to-one and thus invertible.

#### (vi) \( y = \sec^{-1} x \)
- Domain: \((-\infty, -1] \cup [1, \infty)\)
- The secant function, \( \sec x \), is the reciprocal of the cosine function, \( \cos x \). Since \( \cos x \) ranges from \([-1, 1]\), \( \sec x \) is defined for \( x \leq -1 \) or \( x \geq 1 \).
- Range: \([0, \pi] \setminus \{\frac{\pi}{2}\}\)
- The inverse secant function, \( \sec^{-1} x \), returns an angle \( y \) such that \( \sec y = x \) and \( y \) lies in the interval \([0, \pi]\) excluding \( y = \frac{\pi}{2} \). This interval ensures that the function is one-to-one and thus invertible.

Final Answer:

The table correctly lists the domains and ranges for the given inverse trigonometric functions. The solution is verified as follows:

\[
\boxed{
\begin{array}{|c|c|c|c|}
\hline
\text{S.No.} & \text{Function} & \text{Domain} & \text{Range} \\
\hline
(i) & y = \sin^{-1} x & -1 \leq x \leq 1 & -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \\
(ii) & y = \cos^{-1} x & -1 \leq x \leq 1 & 0 \leq y \leq \pi \\
(iii) & y = \tan^{-1} x & x \in \mathbb{R} & -\frac{\pi}{2} < y < \frac{\pi}{2} \\
(iv) & y = \cot^{-1} x & x \in \mathbb{R} & 0 < y < \pi \\
(v) & y = \csc^{-1} x & x \leq -1 \text{ or } x \geq 1 & -\frac{\pi}{2} \leq y \leq \frac{\pi}{2}, \, y \neq 0 \\
(vi) & y = \sec^{-1} x & x \leq -1 \text{ or } x \geq 1 & 0 \leq y \leq \pi, \, y \neq \frac{\pi}{2} \\
\hline
\end{array}
}
\]