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Differential calculus is a subfield of calculus concerned with the study of how things change and is primarily focused on the concept of the derivative. Derivatives describe the rate at which a quantity changes, and understanding how to calculate them is essential for solving real-world problems involving motion, growth, and decay.

In the context of inequalities and functions, differential calculus is used to find the intervals where a function is increasing or decreasing. It helps in sketching the graph of functions and thus makes it easier to solve inequalities. Additionally, knowledge of calculus is necessary to manipulate and understand trigonometric functions and their inverses, including how their derivatives are taken. Learning the derivatives of inverse trigonometric functions, for example, is important as they appear frequently in calculus problems.

Inverse trigonometric functions provide the angle corresponding to a given trigonometric ratio. For example, \(\sin^{-1}(x)\), commonly referred to as arcsin(x), gives you the angle whose sine is x. These functions are particularly useful when you're dealing with trigonometric equations in calculus.

To solve our original exercise, the value of \(\sin^{-1}(\sin 2)\) is needed. Here's where a deeper understanding of these functions becomes vital. Since the sine function is periodic, \(\sin^{-1}(\sin 2)\) finds the angle in the range \[ -\frac{\pi}{2}, \frac{\pi}{2} \] that has the same sine value as 2. This specific knowledge about the properties of inverse trigonometric functions is crucial in determining the correct step in the solution process.

Quadratic inequalities involve an inequality with a quadratic expression, typically in the form \(ax^2 + bx + c > 0\) or \(ax^2 + bx + c < 0\), where a, b, and c are constants. These inequalities can be solved by a variety of methods, such as graphing, using the quadratic formula, or factoring when possible.

In the solution provided, the inequality \(x^2-3x<2-\frac{\pi}{2}\) needed to be rewritten in a factorable form to find the solution set. Recognizing the patterns that lead to factorable expressions and understanding how to interpret these solutions in the context of an inequality are key to solving quadratic inequalities. While it may take practice to master these techniques, they're powerful tools in a mathematician's arsenal for tackling a range of problems.