1. Let's start by stating the problem: You want to learn about the inverse trigonometric functions arcsin, arccos, and arctan. 2. These functions are the inverses of the sine, cosine, and tangent functions respectively. They allow us to find the angle when the value of the sine, cosine, or tangent is known. 3. The notation is: $\arcsin(x)$, $\arccos(x)$, and $\arctan(x)$. 4. Important domains and ranges: - $\arcsin(x)$ is defined for $x \in [-1,1]$ and its range is $[-\frac{\pi}{2}, \frac{\pi}{2}]$. - $\arccos(x)$ is defined for $x \in [-1,1]$ and its range is $[0, \pi]$. - $\arctan(x)$ is defined for all real $x$ and its range is $(-\frac{\pi}{2}, \frac{\pi}{2})$. 5. Formulas: - If $y = \arcsin(x)$ then $\sin(y) = x$. - If $y = \arccos(x)$ then $\cos(y) = x$. - If $y = \arctan(x)$ then $\tan(y) = x$. 6. Example: Find $\arcsin(\frac{1}{2})$. - We look for an angle $y$ such that $\sin(y) = \frac{1}{2}$. - From the unit circle, $y = \frac{\pi}{6}$. - So, $\arcsin(\frac{1}{2}) = \frac{\pi}{6}$. 7. Similarly, $\arccos(\frac{1}{2}) = \frac{\pi}{3}$ because $\cos(\frac{\pi}{3}) = \frac{1}{2}$. 8. And $\arctan(1) = \frac{\pi}{4}$ because $\tan(\frac{\pi}{4}) = 1$. 9. These functions are useful in solving equations involving trigonometric functions and in geometry, physics, and engineering. 10. Remember that these inverse functions return principal values within their specified ranges to ensure they are functions (pass the vertical line test). This overview covers the basics of arcsin, arccos, and arctan functions.