1. The problem is to understand how to solve inverse trigonometry problems, which involve finding an angle when given a trigonometric value. 2. The main inverse trigonometric functions are $\sin^{-1}(x)$, $\cos^{-1}(x)$, and $\tan^{-1}(x)$, which give angles whose sine, cosine, or tangent is $x$ respectively. 3. Important rules: - The domain of $\sin^{-1}(x)$ and $\cos^{-1}(x)$ is $[-1,1]$. - The range of $\sin^{-1}(x)$ is $[-\frac{\pi}{2}, \frac{\pi}{2}]$. - The range of $\cos^{-1}(x)$ is $[0, \pi]$. - The range of $\tan^{-1}(x)$ is $(-\frac{\pi}{2}, \frac{\pi}{2})$. 4. To solve an inverse trig problem, identify the function and value, then use the inverse function to find the angle. Example: Solve $\sin^{-1}(\frac{1}{2})$. 5. Using the definition, $\sin^{-1}(\frac{1}{2})$ is the angle $\theta$ such that $\sin(\theta) = \frac{1}{2}$. 6. From the unit circle, $\sin(\frac{\pi}{6}) = \frac{1}{2}$, and since $\frac{\pi}{6}$ is in the range of $\sin^{-1}$, the solution is $\theta = \frac{\pi}{6}$. 7. Always check the domain and range to ensure the solution is valid. This method applies similarly for $\cos^{-1}$ and $\tan^{-1}$ problems.