To solve trigonometric inequalities effectively, you can use some simple steps that make the process easier to understand. Here’s how to do it:

• Identify the Function: Start by figuring out the trigonometric function you are dealing with. This could be $\sin(x)$, $\cos(x)$, or $\tan(x)$. Knowing which function it is helps you understand how it behaves over time because these functions repeat themselves.

• Set Up the Inequality: Rewrite the inequality in a clear format. For example, if you have $\sin(x) > 0$, note where the function equals zero. For $\sin(x)$, that happens at points $x = n\pi$, where $n$ is any whole number.

• Determine Key Values: Find the important points by solving the equation $f(x) = 0$. These points break the number line into sections where the function is either positive or negative. For $\sin(x) > 0$, the critical points are $0, \pi, 2\pi$, and so on.

• Test Intervals: Pick a test point from each section created by the critical points. By plugging these test points into $f(x)$, you can see whether the function is positive or negative in those sections.

• Combine Information: Gather your results from testing and create a sign chart, which is a visual aid. This will help you see which sections satisfy the inequality.

• State the Solution: Finally, explain your results using interval notation. For example, if $\sin(x) > 0$ in the sections $(0, \pi)$ and $(2\pi, 3\pi)$, you can write the solution like this: $x \in (0, \pi) \cup (2\pi, 3\pi)$.

• Consider Domain Restrictions: Make sure to think about any limits based on the specific problem. For example, the values of $x$ might be limited to $[0, 2\pi)$ or some other range.

By following these steps, solving trigonometric inequalities becomes a clear and organized process. This makes it easier to find solutions and understand what you are doing.