1. The Mirror Effect ($y = x$)

The graph of any inverse function $f^{-1}(x)$ is simply the geometric reflection of the original function $f(x)$ across the diagonal line $y = x$.

If a point $(a, b)$ exists on the standard trigonometric curve, the point $(b, a)$ will exist on the inverse trigonometric curve. Because of this swap, the domain of the original function becomes the range of the inverse, and vice versa.

2. Bounded Curves: $\sin^{-1}(x)$ and $\cos^{-1}(x)$

Because standard sine and cosine waves naturally oscillate between $-1$ and $1$, their inverse graphs are tightly bounded horizontally. They only exist for $x$-values between $[-1, 1]$.

• $\sin^{-1}(x)$: An odd, strictly increasing curve crossing exactly through the origin $(0,0)$. It rises from $( -1, -\pi/2 )$ up to $( 1, \pi/2 )$.
• $\cos^{-1}(x)$: A strictly decreasing curve. It starts high at $( -1, \pi )$, crosses the y-axis at $(0, \pi/2)$, and ends at $(1, 0)$.

3. Unbounded Domains & Approaching Limits

The standard tangent and cotangent functions stretch upward and downward indefinitely, bounded by invisible vertical lines they never touch. When reflected across $y=x$, these vertical bounds turn into Horizontal Boundaries for $\tan^{-1}(x)$ and $\cot^{-1}(x)$. Their domains are infinitely wide ($\mathbb{R}$).

4. Split Branches: $\sec^{-1}(x)$ and $\csc^{-1}(x)$

Since the standard secant and cosecant graphs never enter the region between $y=-1$ and $y=1$, their inverses have a massive "gap" in their domain. They do not exist for $x \in (-1, 1)$.

Because of this gap, their graphs are split into two completely separate curved branches. One branch heads towards positive infinity, and the other towards negative infinity, both flattening out along a horizontal line in the middle.

• $\csc^{-1}(x)$: Has a horizontal boundary line exactly on the x-axis ($y = 0$) which it approaches but never touches.
• $\sec^{-1}(x)$: Has a horizontal boundary line exactly halfway up the graph at $y = \pi/2$ which it never crosses.