Q.State True or False: The graph of inverse trigonometric function can be obtained from the graph of their corresponding trigonometric function by interchanging $x$ and $y$ axes.

Concept: Inverse Trigonometric Graphs — The graph of an inverse function is the reflection of the original function's graph across the line $y = x$, which is equivalent to swapping the $x$ and $y$ axes.

Reasoning:

1. If $y = f(x)$ is a trigonometric function, its inverse $f^{-1}(x)$ satisfies $f^{-1}(y) = x$ whenever $y = f(x)$.
2. This means every point $(a, b)$ on the graph of $f$ corresponds to the point $(b, a)$ on the graph of $f^{-1}$. …

Why This Works — The Core Idea

When you have a function $y = f(x)$, its inverse $f^{-1}(x)$ is defined by swapping the roles of input and output. If a point $(a, b)$ lies on the graph of $f$, then $(b, a)$ lies on the graph of $f^{-1}$. Geometrically, this swap is exactly a reflection across the line $y = x$.

For trigonometric functions, the same logic holds — but with one important caveat: trigonometric functions are not one-to-one over their entire domain. So we restrict them to a principal branch (e.g., $\sin x$ on $[-\pi/2, \pi/2]$) to define a proper inverse. Once that restriction is in place, the graph of the inverse is indeed the mirror image of that restricted graph across $y = x$.

Step-by-Step Reasoning

1. Recall the definition of an inverse function.

   If $y = f(x)$ is one-to-one, then $f^{-1}(y) = x$ exactly when $f(x) = y$. This means the ordered pairs are swapped: $(x, y)$ on $f$ becomes $(y, x)$ on $f^{-1}$.

2. What does "interchanging $x$ and $y$ axes" mean?

   In coordinate geometry, swapping the axes means that the horizontal axis now represents the old $y$-values, and the vertical axis represents the old $x$-values. Plotting $(y, x)$ for every point $(x, y)$ on the original curve is precisely the reflection across $y = x$.

3. Apply this to trigonometric functions.

   Take $y = \sin x$ restricted to $x \in [-\pi/2, \pi/2]$. Its inverse is $y = \sin^{-1} x$ (or $\arcsin x$).

   • A point like $(\pi/6, 1/2)$ on $\sin x$ becomes $(1/2, \pi/6)$ on $\sin^{-1} x$.
   • The entire curve of $\sin^{-1} x$ is the mirror image of the restricted sine curve across $y = x$.

4. The same holds for all six inverse trigonometric functions. …

Method: Relate an inverse's graph to the original

Steps

Step 1: Use the point-swap property.

If $(a,b)$ lies on $f$, then $(b,a)$ lies on $f^{-1}$ — input and output swap roles.

Step 2: Interpret the swap geometrically.

Plotting $(b,a)$ for every point $(a,b)$ is a reflection across the line $y=x$, i.e. interchanging the $x$- and $y$-axes. …

Common Mistakes

Mistake 1: Applying the reflection to the full, unrestricted trig graph.

Why it's wrong: a full sine or cosine curve fails the horizontal line test, so its reflection across $y=x$ is not a function. Correct approach: reflect the restricted (principal-branch) curve. …