Trigonometric Equations: Inequalities
Trigonometric Equations: Inequalities
Trigonometric inequalities are mathematical expressions involving trigonometric functions such as sine, cosine, tangent, etc., and an inequality sign (>, <, ≥, or ≤). Solving these inequalities often involves finding the range of angles that satisfy the given condition.
Understanding Trigonometric Functions
Before delving into inequalities, it is essential to have a firm grasp of the basic trigonometric functions and their properties:
- Sine (sin): A function that gives the ratio of the opposite side to the hypotenuse in a right-angled triangle.
- Cosine (cos): A function that gives the ratio of the adjacent side to the hypotenuse.
- Tangent (tan): A function that gives the ratio of the opposite side to the adjacent side.
These functions have specific ranges and periods that are important when solving inequalities:
| Function | Range | Period |
|---|---|---|
| sin(x) | [-1, 1] | 2π |
| cos(x) | [-1, 1] | 2π |
| tan(x) | (-∞, ∞) | π |
Solving Trigonometric Inequalities
To solve trigonometric inequalities, follow these general steps:
- Isolate the Trigonometric Function: Try to express the inequality in terms of a single trigonometric function.
- Use Trigonometric Identities: Apply identities to simplify the inequality if necessary.
- Find the Reference Angle: Determine the angle that corresponds to the trigonometric value in the inequality.
- Determine the Solution Intervals: Use the unit circle or trigonometric graphs to find the range of angles that satisfy the inequality.
- Express the Solution: Write the solution in terms of intervals or using periodicity properties.
Example 1: Basic Inequality
Solve the inequality $ \sin(x) > \frac{1}{2} $.
- Isolate the Trigonometric Function: The function is already isolated.
- Use Trigonometric Identities: No identities are needed.
- Find the Reference Angle: The reference angle for $ \sin^{-1}(\frac{1}{2}) $ is $ \frac{\pi}{6} $.
- Determine the Solution Intervals: Since the sine function is positive in the first and second quadrants, the solution intervals are $ x \in (\frac{\pi}{6}, \frac{5\pi}{6}) + 2k\pi $, where $ k $ is an integer.
- Express the Solution: The solution is $ x \in (\frac{\pi}{6} + 2k\pi, \frac{5\pi}{6} + 2k\pi) $ for any integer $ k $.
Example 2: Compound Inequality
Solve the inequality $ 0 < \cos(x) \leq \frac{\sqrt{3}}{2} $.
- Isolate the Trigonometric Function: The function is already isolated.
- Use Trigonometric Identities: No identities are needed.
- Find the Reference Angle: The reference angle for $ \cos^{-1}(\frac{\sqrt{3}}{2}) $ is $ \frac{\pi}{6} $.
- Determine the Solution Intervals: The cosine function is positive in the first and fourth quadrants. However, since $ 0 < \cos(x) $, we exclude $ x = 2k\pi $. The solution intervals are $ x \in (0, \frac{\pi}{6}) \cup (\frac{11\pi}{6}, 2\pi) + 2k\pi $.
- Express the Solution: The solution is $ x \in (2k\pi, \frac{\pi}{6} + 2k\pi) \cup (\frac{11\pi}{6} + 2k\pi, 2\pi + 2k\pi) $ for any integer $ k $.
Important Points to Remember
- Periodicity: Trigonometric functions repeat their values at regular intervals, known as their periods.
- Symmetry: Sine and tangent are odd functions ($ f(-x) = -f(x) $), while cosine is an even function ($ f(-x) = f(x) $).
- Unit Circle: The unit circle is a helpful tool for visualizing the values of trigonometric functions at various angles.
- Graphs: The graphs of sine, cosine, and tangent functions can provide a visual method for determining the solution set of an inequality.
Conclusion
Solving trigonometric inequalities requires a combination of algebraic manipulation, understanding of trigonometric functions and their properties, and sometimes visual aids such as graphs or the unit circle. Practice with various types of inequalities will improve problem-solving skills in this area, which is essential for success in exams involving trigonometry.