Inverse trigonometric functions are helpful for finding angles when we know the side ratios of a triangle. The main ones are:

• $\sin^{-1}(x)$
• $\cos^{-1}(x)$
• $\tan^{-1}(x)$

These functions are really useful in fields like engineering, physics, and computer graphics.

Here’s how to graph these functions step by step:

1. Identify the Range and Domain:

   • For $\sin^{-1}(x)$:
     • Domain: $[-1, 1]$
     • Range: $[-\frac{\pi}{2}, \frac{\pi}{2}]$
   • For $\cos^{-1}(x)$:
     • Domain: $[-1, 1]$
     • Range: $[0, \pi]$
   • For $\tan^{-1}(x)$:
     • Domain: All real numbers
     • Range: $(-\frac{\pi}{2}, \frac{\pi}{2})$

2. Plot Points:

   • Choose values for $x$ that fit the domains above.
   • Calculate the $y$ values that go with those $x$ values.

3. Connect the Dots:

   • Draw a smooth line through your points.

Each graph looks a bit different:

• The graph of $\sin^{-1}(x)$ goes up from $(-1, -\frac{\pi}{2})$ to $(1, \frac{\pi}{2})$.
• The graph of $\cos^{-1}(x)$ goes down from $(1, 0)$ to $(-1, \pi)$.
• The graph of $\tan^{-1}(x)$ gets closer to $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ as $x$ moves toward $-\infty$ and $\infty$.

Knowing how to read these graphs can help you understand their values! For instance, if you want to find $\sin^{-1}(0.5)$, you are looking for the angle that has a sine value of $0.5$. This angle is $\frac{\pi}{6}$ or $30^\circ$.