Question

Assertion(A): All trigonometric functions have their inverses over the respective principal domain.

Reason(R): The inverse of cos$^{-1}$ x: [-1,1]→ [0, $\pi$] exists.

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true but R is not the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Answer 1

Answer: (B)

(b) Both A and R are true but R is not the correct explanation of A

Answer 2

Explanation of the solution:

1. Analyze Assertion (A): "All trigonometric functions have their inverses over the respective principal domain."

   • Trigonometric functions (sin, cos, tan, etc.) are periodic and thus not one-to-one over their natural domains.
   • To define their inverse functions, their domains are restricted to specific intervals (principal domains) where they become one-to-one and onto their ranges.
   • For example, $\sin x$ restricted to $[-\pi/2, \pi/2]$ is one-to-one and onto $[-1, 1]$, allowing $\sin^{-1} x$ to be defined from $[-1, 1]$ to $[-\pi/2, \pi/2]$. Similar restrictions apply to all other trigonometric functions.
   • Therefore, Assertion (A) is True.

2. Analyze Reason (R): "The inverse of cos$^{-1}$ x: [-1,1]→ [0, $\pi$] exists."

   • Let $f(x) = \cos^{-1} x$. This function is defined with a domain of $[-1, 1]$ and a range of $[0, \pi]$.
   • By definition, $\cos^{-1} x$ is the inverse of the cosine function when the cosine function's domain is restricted to $[0, \pi]$. This means $\cos^{-1} x$ is a bijection from $[-1, 1]$ to $[0, \pi]$.
   • Any function that is a bijection (one-to-one and onto) has an inverse.
   • The inverse of $f(x) = \cos^{-1} x$ is $f^{-1}(y) = \cos y$, with domain $[0, \pi]$ and range $[-1, 1]$.
   • Therefore, Reason (R) is True.

3. Evaluate if R is the correct explanation of A:

   • Assertion (A) makes a general statement about all trigonometric functions and the necessity of principal domains for their inverses to exist.
   • Reason (R) states a specific fact about the existence of the inverse of $\cos^{-1} x$. The existence of the inverse of $\cos^{-1} x$ (which is $\cos x$ restricted to its principal domain) does not explain why the original trigonometric functions need domain restrictions to have inverses. It's a consequence of the definition of inverse trigonometric functions, not an explanation for the general principle.
   • Thus, R is not the correct explanation for A.

Conclusion: Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation for Assertion (A).