Analyzing Domain-Specific Inequalities

Introduction

Key Concepts

1. Understanding Domain in Trigonometric Inequalities

In the context of trigonometric inequalities, the domain refers to the set of all possible input values (angles) for which the trigonometric functions are defined. Identifying the correct domain is crucial as it ensures the validity of the solutions obtained. For instance, the sine and cosine functions are defined for all real numbers, but their ranges are limited to [-1, 1].

2. Types of Trigonometric Inequalities

• Linear Trigonometric Inequalities: Inequalities where the trigonometric function appears to the first power, such as $ \sin(x) > \frac{1}{2} $.
• Quadratic Trigonometric Inequalities: Involves squared trigonometric functions, for example, $ \cos^2(x) - \cos(x) - 2 < 0 $.
• Compound Trigonometric Inequalities: Combines multiple trigonometric expressions, like $ \sin(x) \cos(x) \geq \frac{1}{4} $.

3. Solving Linear Trigonometric Inequalities

To solve linear trigonometric inequalities, follow these steps:

1. Isolate the Trigonometric Function: Begin by isolating the trigonometric function on one side of the inequality.
2. Determine the Critical Points: Find the values of the variable where the trigonometric function equals the boundary value.
3. Analyze the Intervals: Divide the domain into intervals based on the critical points and test each interval to determine where the inequality holds.
4. Express the Solution: Present the solution in interval notation or using inequalities.

Example: Solve $ \sin(x) > \frac{1}{2} $ for $ 0 < x < 2\pi $.

Solution:
The inequality $ \sin(x) > \frac{1}{2} $ holds true where the sine function is above $ \frac{1}{2} $. This occurs in the intervals:

Thus, the solution is $ x \in \left(\frac{\pi}{6}, \frac{5\pi}{6}\right) $.

4. Solving Quadratic Trigonometric Inequalities

Quadratic trigonometric inequalities involve higher powers of trigonometric functions. The approach is similar to solving quadratic equations:

1. Let $ u = \cos(x) $: Substitute to simplify the inequality.
2. Solve the Quadratic Inequality: Find the values of $ u $ that satisfy the inequality.
3. Back-Substitute: Replace $ u $ with $ \cos(x) $ and solve for $ x $ within the given domain.

Example: Solve $ \cos^2(x) - \cos(x) - 2 < 0 $.

Solution:
Let $ u = \cos(x) $. The inequality becomes: $$ u^2 - u - 2 < 0 $$ Factoring: $$ (u - 2)(u + 1) < 0 $$ Critical points are $ u = 2 $ and $ u = -1 $. Testing intervals:

• For $ u < -1 $, the expression is positive.
• For $ -1 < u < 2 $, the expression is negative.
• For $ u > 2 $, the expression is positive.

Since $ u = \cos(x) $ must satisfy $ -1 \leq u \leq 1 $, the solution is $ -1 < \cos(x) < 2 $, which simplifies to $ \cos(x) > -1 $. This holds for all $ x $ except where $ \cos(x) = -1 $, i.e., $ x \neq \pi + 2k\pi $ for any integer $ k $.

5. Solving Compound Trigonometric Inequalities

Compound inequalities involve multiple trigonometric expressions and may require the use of identities or substitutions to simplify:

1. Simplify the Inequality: Use trigonometric identities to combine or reduce terms.
2. Find Common Solutions: Determine the intersection of solutions from each part of the compound inequality.
3. Verify Solutions: Ensure that the solutions satisfy all parts of the inequality.

Example: Solve $ \sin(x) \cos(x) \geq \frac{1}{4} $.

Solution:
Using the identity $ 2\sin(x)\cos(x) = \sin(2x) $, the inequality becomes: $$ \sin(2x) \geq \frac{1}{2} $$ Solving for $ 2x $: $$ 2x \in \left[\frac{\pi}{6}, \frac{5\pi}{6}\right] + 2k\pi \quad \text{for any integer } k $$ Dividing by 2: $$ x \in \left[\frac{\pi}{12}, \frac{5\pi}{12}\right] + k\pi \quad \text{for any integer } k $$

6. Graphical Interpretation of Trigonometric Inequalities

Graphing trigonometric functions provides a visual understanding of where inequalities hold:

• Identify Intersection Points: Points where the trigonometric function intersects the boundary value.
• Determine Sign Changes: Analyze where the function is above or below the boundary.
• Shade Relevant Regions: Highlight the intervals on the graph that satisfy the inequality.

Example: Graphically solve $ \sin(x) > \frac{1}{2} $.

The graph of $ \sin(x) $ intersects $ y = \frac{1}{2} $ at $ x = \frac{\pi}{6} $ and $ x = \frac{5\pi}{6} $ within $ 0 < x < 2\pi $. The sine function is above $ \frac{1}{2} $ in the interval $ \left(\frac{\pi}{6}, \frac{5\pi}{6}\right) $.

7. Utilizing Unit Circle for Solving Inequalities

The unit circle is a valuable tool for solving trigonometric inequalities by providing precise angle measures where functions attain specific values:

• Select the Relevant Quadrants: Determine in which quadrants the trigonometric function satisfies the inequality.
• Calculate Reference Angles: Find the acute angles corresponding to the boundary values.
• Express Solutions in Terms of $ \pi $: Provide solutions using radians for standardization.

Example: Solve $ \tan(x) \leq 1 $ for $ 0 < x < 2\pi $.

Solution:
$ \tan(x) = 1 $ at $ x = \frac{\pi}{4} $ and $ x = \frac{5\pi}{4} $. The tangent function is less than or equal to 1 in the intervals: $$ (0, \frac{\pi}{4}] \quad \text{and} \quad \left(\frac{5\pi}{4}, 2\pi\right) $$

8. Applying Inverse Trigonometric Functions

Inverse trigonometric functions are used to find angle measures corresponding to specific function values, essential in solving inequalities:

• Use Principal Values: Apply the correct range for principal values to ensure accurate solutions.
• Extend Solutions Periodically: Account for the periodic nature of trigonometric functions by adding multiples of the period.

Example: Solve $ \cos^{-1}\left(\frac{x}{2}\right) > \frac{\pi}{3} $.

Solution:
First, solve $ \cos^{-1}\left(\frac{x}{2}\right) > \frac{\pi}{3} $ which implies: $$ \frac{x}{2} < \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} $$ Thus, $$ x < 1 $$ Considering the domain of $ \cos^{-1} $, which requires $ -1 \leq \frac{x}{2} \leq 1 $, we have: $$ -2 \leq x \leq 2 $$ Combining the inequalities: $$ -2 \leq x < 1 $$

9. Addressing Special Cases and Restrictions

Certain trigonometric functions have inherent restrictions that must be considered when solving inequalities:

• Division by Zero: Avoid values that make the denominator zero in rational expressions.
• Undefined Functions: Ensure that the arguments of inverse functions lie within their domain.

Example: Solve $ \frac{\sin(x)}{\cos(x)} \geq 1 $.

Solution:
Rewrite as: $$ \tan(x) \geq 1 $$ $ \tan(x) $ is undefined at $ x = \frac{\pi}{2} + k\pi $ for any integer $ k $. The solution within $ 0 < x < 2\pi $ is: $$ \left[\frac{\pi}{4}, \frac{\pi}{2}\right) \cup \left[\frac{5\pi}{4}, \frac{3\pi}{2}\right) $$

10. Verifying Solutions

Always verify solutions by substituting them back into the original inequality to ensure they satisfy all conditions:

• Check Boundary Points: Ensure that equality holds where applicable.
• Confirm Within Domain: Solutions must lie within the defined domain of the problem.

Example: Verify the solution $ x \in \left(\frac{\pi}{6}, \frac{5\pi}{6}\right) $ for $ \sin(x) > \frac{1}{2} $.

Testing $ x = \frac{\pi}{4} $: $$ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \approx 0.707 > 0.5 \quad \text{(Valid)} $$ Testing $ x = \frac{\pi}{3} $: $$ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \approx 0.866 > 0.5 \quad \text{(Valid)} $$ Both tests confirm the solution is correct.

Comparison Table

Summary and Key Takeaways