91Ó°ÊÓ

The Pythagorean trigonometric identity is a fundamental relation in trigonometry that relates the squares of the sine and cosine of an angle. The identity is expressed as \( \sin^2{x} + \cos^2{x} = 1 \). This relationship is derived from the Pythagorean theorem and holds true for all angles x.

Understanding this identity is crucial when solving trigonometric equations and inequalities because it allows us to express \( \sin^2{x} \) or \( \cos^2{x} \) solely in terms of the other, simplifying the equation. In the given exercise, for instance, \( \cos^2{x} \) was substituted using \( 1 - \sin^2{x} \) to eliminate the cosine term, transforming the inequality into a more familiar quadratic form. This transformation is a key step that opens the door to further techniques in algebra, such as solving quadratic inequalities.

Quadratic inequalities are algebraic expressions that relate a quadratic polynomial to a zero or another polynomial through an inequality sign (>, <, \(\geq\), or \(\leq\)). To solve these inequalities, one typically moves all terms to one side of the inequality to obtain a standard form similar to a quadratic equation: \(ax^2 + bx + c > 0\), as seen in the exercise.

Once this form is achieved, we can solve the corresponding quadratic equation \(ax^2 + bx + c = 0\) to find the roots or critical values. These critical values partition the number line into intervals on which the quadratic polynomial maintains a consistent sign. By testing points from each interval, or analyzing the vertex and direction of the parabola, we can determine where the inequality holds true. This step is essential for accurately identifying the solution set of the inequality.

Inverse trigonometric functions allow us to find the angle that corresponds to a specific trigonometric value. They are the inverses of the trigonometric functions and include the arcsine (\(\arcsin\)), arccosine (\(\arccos\)), and arctangent (\(\arctan\)), among others.

In solving trigonometric inequalities, once we have simplified the inequality to an expression involving a single trigonometric function, such as \(\sin(x)\), we can employ the inverse function to find the angle x. It's important to remember that trigonometric functions are periodic and not one-to-one unless restricted to proper intervals. Therefore, the solutions will typically involve adding integer multiples of the function's period to the principal value obtained from the inverse function. In the context of the exercise, after solving for \(z = \sin(x)\), the use of the \(\arcsin\) function helps in unwrapping the substitution and determining the angles that satisfy the original inequality.