1. The problem involves solving a trigonometric equation where the solution $\frac{7}{12}\pi$ was found. 2. Your method is to isolate the trigonometric function (e.g., $\sin x$) and then apply the inverse function to both sides. 3. For sine equations, the general solutions are $x = \sin^{-1}(y)$ and $x = \pi - \sin^{-1}(y)$, where $y$ is the value of the sine function. 4. Step-by-step, if you have $\sin x = a$, then: - First solution: $x = \sin^{-1}(a)$ - Second solution: $x = \pi - \sin^{-1}(a)$ 5. For example, if $\sin x = \frac{\sqrt{3}}{2}$, then: - $x = \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}$ - $x = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$ 6. If your solution was $\frac{7}{12}\pi$, it likely came from applying $x = \pi + \sin^{-1}(a)$ or $x = 180^\circ + \sin^{-1}(a)$ in degrees, which corresponds to the general solution for angles beyond $\pi$ in sine equations. 7. So, your approach is correct: isolate the trig function, apply inverse, then use the general solution formulas to find all possible angles. 8. This explains how $\frac{7}{12}\pi$ can be obtained using your method. Final answer: $x = \frac{7}{12}\pi$ is one of the solutions obtained by $x = \pi - \sin^{-1}(a)$ or $x = \pi + \sin^{-1}(a)$ depending on the problem context.