91Ó°ÊÓ

Understanding quadratic inequalities is crucial for solving many real-world problems and equations that involve a squared term. A quadratic inequality is generally of the form \(ax^2 + bx + c < 0\), \(ax^2 + bx + c \leq 0\), \(ax^2 + bx + c > 0\), or \(ax^2 + bx + c \geq 0\). Solving these requires finding the roots of the corresponding quadratic equation \(ax^2 + bx + c = 0\).

To solve a quadratic inequality, follow these steps:

• Find the roots of the quadratic equation using factoring, completing the square, or the quadratic formula.
• Determine the intervals on the number line that satisfy the inequality. This is done by selecting test values from the intervals formed by the roots.
• Analyze the sign of the expression within each interval to understand which intervals satisfy the inequality.

In the original solution, the quadratic inequality \((x - 2)(x + 1) < 0\) was solved by using the roots \(x = -1\) and \(x = 2\). By testing intervals such as \((-\infty, -1)\), \((-1, 2)\), and \((2, \infty)\), we determined \((-1, 2)\) satisfies the inequality.

Trigonometric identities are equalities that involve trigonometric functions and hold true for all values of the involved variables. These identities are essential for simplifying and solving trigonometric equations and expressions.

Some fundamental trigonometric identities include:

• \(\sin^2\theta + \cos^2\theta = 1\)
• \(1 + \tan^2\theta = \sec^2\theta\)
• \(1 + \cot^2\theta = \csc^2\theta\)
• Sum and difference identities, like \(\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b\).
• Double angle formulas, such as \(\sin 2\theta = 2 \sin \theta \cos \theta\).

In solving trigonometric inequalities, recognizing patterns and substituting identities can simplify complex expressions. In the exercise, rewriting \(\frac{\sin^3 \theta - \cos^3 \theta}{\sin \theta - \cos \theta}\) using the identity \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\) demonstrates how identities help simplify and solve problems.

Radian measure is an alternative to measuring angles in degrees, using the radius of a circle. One radian is the angle formed when the radius forms an arc length equal to the radius itself. This connection provides a natural way to define angles based on circular motion.

Key points about radian measure:

• There are \(2\pi\) radians in a full circle, which corresponds to \(360^\circ\).
• \(\pi\) radians equals \(180^\circ\). Thus \(\frac{\pi}{4}\) radians is equivalent to \(45^\circ\).
• Radians are often favored in mathematics and physics because they simplify the relationship between angles and arc lengths.

Brushing up on converting between degrees and radians is important. Conversion is simple: Multiply degrees by \(\frac{\pi}{180}\) to get radians, and multiply radians by \(\frac{180}{\pi}\) for degrees. Understanding angles in radians is essential for solving problems involving circular or periodic motion, as seen in trigonometric equations within the given solution.