Trigonometric inequalities

Trigonometric Inequalities

Trigonometric inequalities are inequalities that involve trigonometric functions such as sine, cosine, tangent, etc. Solving these inequalities often requires a good understanding of the properties of trigonometric functions and their graphs. In this guide, we will explore the concept of trigonometric inequalities, how to solve them, and some important points to remember.

Basic Trigonometric Functions

Before we delve into inequalities, let's briefly review the basic trigonometric functions and their ranges:

Function Definition (right-angled triangle) Range
$\sin(\theta)$ Opposite side / Hypotenuse $[-1, 1]$
$\cos(\theta)$ Adjacent side / Hypotenuse $[-1, 1]$
$\tan(\theta)$ Opposite side / Adjacent side $(-\infty, \infty)$

Solving Trigonometric Inequalities

To solve trigonometric inequalities, we often follow these general steps:

  1. Identify the Interval: Determine the interval over which you need to solve the inequality (e.g., $[0, 2\pi]$ or $[0, \pi]$).
  2. Isolate the Trigonometric Function: Try to isolate the trigonometric function on one side of the inequality.
  3. Use Trigonometric Identities: Apply trigonometric identities to simplify the inequality if necessary.
  4. Find Critical Points: Determine the values of the variable where the trigonometric function equals the boundary of the inequality.
  5. Test Intervals: Test the intervals between the critical points to see where the inequality holds true.
  6. Combine Solutions: Combine the intervals that satisfy the inequality to find the solution set.

Examples

Let's look at some examples to illustrate how to solve trigonometric inequalities.

Example 1: Solving a Basic Sine Inequality

Solve the inequality $\sin(x) > \frac{1}{2}$ for $x$ in the interval $[0, 2\pi]$.

Solution:

  1. The interval is already given as $[0, 2\pi]$.
  2. The trigonometric function $\sin(x)$ is already isolated.
  3. No identities are needed to simplify this inequality.
  4. We find the critical points where $\sin(x) = \frac{1}{2}$. This occurs at $x = \frac{\pi}{6}$ and $x = \frac{5\pi}{6}$ within the given interval.
  5. We test the intervals between the critical points:
    • For $x$ in $(0, \frac{\pi}{6})$, $\sin(x) < \frac{1}{2}$.
    • For $x$ in $(\frac{\pi}{6}, \frac{5\pi}{6})$, $\sin(x) > \frac{1}{2}$.
    • For $x$ in $(\frac{5\pi}{6}, 2\pi)$, $\sin(x) < \frac{1}{2}$.
  6. The solution set is $(\frac{\pi}{6}, \frac{5\pi}{6})$.

Example 2: Solving a Cosine Inequality with Identities

Solve the inequality $2\cos^2(x) - 1 < 0$ for $x$ in the interval $[0, 2\pi]$.

Solution:

  1. The interval is $[0, 2\pi]$.
  2. The trigonometric function $\cos(x)$ is not isolated, but we can use an identity.
  3. Apply the double-angle identity: $2\cos^2(x) - 1 = \cos(2x)$.
  4. Find the critical points where $\cos(2x) = 0$. This occurs at $x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$ within the given interval.
  5. Test the intervals between the critical points:
    • For $x$ in $(0, \frac{\pi}{4})$, $\cos(2x) > 0$.
    • For $x$ in $(\frac{\pi}{4}, \frac{3\pi}{4})$, $\cos(2x) < 0$.
    • For $x$ in $(\frac{3\pi}{4}, \frac{5\pi}{4})$, $\cos(2x) > 0$.
    • For $x$ in $(\frac{5\pi}{4}, \frac{7\pi}{4})$, $\cos(2x) < 0$.
    • For $x$ in $(\frac{7\pi}{4}, 2\pi)$, $\cos(2x) > 0$.
  6. The solution set is $(\frac{\pi}{4}, \frac{3\pi}{4}) \cup (\frac{5\pi}{4}, \frac{7\pi}{4})$.

Important Points to Remember

  • Periodicity: Trigonometric functions are periodic, so the solutions to inequalities will often repeat at regular intervals.
  • Symmetry: Use the symmetry properties of trigonometric functions to simplify the problem.
  • Graphical Approach: Sometimes, sketching the graph of the trigonometric function can help visualize the solution.
  • Domain Restrictions: Remember that the domain of $\tan(x)$ and $\cot(x)$ excludes values where the function is undefined (e.g., $\tan(x)$ is undefined at $x = \frac{\pi}{2} + k\pi$, where $k$ is an integer).

By following these steps and keeping these points in mind, you should be able to tackle a wide range of trigonometric inequalities. Practice with various examples to become more comfortable with these types of problems.