91Ó°ÊÓ

In the world of mathematics, quadratic expressions are vital, particularly when dealing with trigonometric inequalities. A quadratic expression usually takes the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. For our original exercise, we encounter a quadratic form through trigonometric manipulation.Initially, we transformed the trigonometric expression \( \cos 2x + 3\sin x \) into a quadratic form by using trigonometric identities. This involved rewriting \( \cos 2x \) as \( 1 - 2\sin^2 x \), thus reshaping the expression into \( 2\sin^2 x - 3\sin x + 1 \). Here, the quadratic expression is pinpointed by realizing its form matches \( ax^2 + bx + c \) with \( a = 2 \), \( b = 3 \), and \( c = 1 \).Quadratic expressions are core to solving inequalities, which we'll explore later. But first, it's crucial to understand the concavity of the parabola represented by the quadratic expression. Since \( a = 2 \) is positive, the parabola opens upwards, indicating any minimum value obtained is a lower bound, aiding in determining the expression's range.

Trigonometric identities are essential tools used to simplify and solve equations involving trigonometric functions. They allow us to express one trigonometric function in terms of another or simplify complex expressions. This often provides clearer insights and simpler forms that are more manageable during problem-solving.In the exercise, we utilized the identity \( \cos 2x = 1 - 2\sin^2 x \) to rewrite \( \cos 2x + 3\sin x \). By expressing \( \cos 2x \) in terms of sine, we avoided dealing directly with two different trigonometric functions simultaneously. Instead, it allowed us to work in one trigonometric space, focusing on \( \sin x \) alone.This transformation is vital not only for rewriting expressions but also for seeing relationships between different parts of the problem that might not be evident at first glance. Trigonometric identities, such as the one used here, offer a bridge between expressions, illuminating hidden paths to solutions and making it easier to solve inequalities.

Solving inequalities, especially those involving trigonometric expressions, requires a strategic approach to finding bounds or ranges that satisfy the inequality. The aim is to isolate the variable and determine the set of values that make the expression true.For the quadratic expression transformed in the problem, we identified its range of values using the vertex formula of quadratics, which gives the minimum or maximum points. By applying \( x = \frac{b}{2a} \), we found a crucial point on the curve. This method provided insight into how the quadratic behaves, revealing that the lowest point (minimum) of \( 2\sin^2 x - 3\sin x + 1 \) occurs at \( \frac{-1}{8} \).Building on that, we used fundamental algebraic techniques to expand the range found, aligning it with the inequality requirements. Multiplying and making transformations helped achieve the final result, proving \(-4 \leq \cos 2x + 3\sin x \leq \frac{17}{8} \). Solving such trigonometric inequalities often necessitates balancing simple algebra with deeper trigonometric understanding to determine all possible values.