Easy Ways to Solve Trigonometric Inequalities

Trigonometric inequalities are math problems that use trigonometric ratios and need a range of answers. Here are some simple steps to solve these inequalities, especially for AS-Level students.

1. Know the Trigonometric Functions

It’s important to be familiar with the graphs of sine, cosine, and tangent functions. Here are some key points:

• Sine and cosine repeat every $2\pi$, while tangent repeats every $\pi$.
• Sine and cosine values range from -1 to 1, but tangent can be much larger or smaller, going from $-\infty$ to $+\infty$.

By understanding these functions, you can find possible solutions for certain ranges.

2. Changing the Inequality

Make the trigonometric inequality easier to work with. You can do this by:

• Using identities like the Pythagorean identity: ( \sin^2(x) + \cos^2(x) = 1 ).
• Applying angle addition or double angle formulas to simplify the functions.

For example, if you have the inequality $2 \sin(x) - 1 < 0$, you can change it to $\sin(x) < \frac{1}{2}$.

3. Finding Important Values

Look for key points where the function reaches its highest and lowest values. For $\sin(x)$ and $\cos(x)$, these points include $0$, $1$, and $-1$. For $\tan(x)$, remember it is undefined at odd multiples of $\frac{\pi}{2}$. Important angles in radians include:

• $0$, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, $\frac{\pi}{2}$, $\pi$, etc.

These angles help you set up ranges to check for solutions.

4. Testing Ranges

Choose test points from the ranges you identified. Here’s how to do it:

• Break the number line into sections based on the important values.
• Pick a test value from each section to see if the inequality is correct.

For instance, if you’re solving $2 \sin(x) < 1$, the key points are $x = \frac{\pi}{6}$ and $x = \frac{5\pi}{6}$. You can test sections like $(0, \frac{\pi}{6})$, $(\frac{\pi}{6}, \frac{5\pi}{6})$, and $(\frac{5\pi}{6}, 2\pi)$.

5. Collecting Answers

Once you’ve tested each section, gather your results to show where the inequality works. For example, you might find that the inequality $2 \sin(x) < 1$ is true in the intervals $(0, \frac{\pi}{6})$ and $(\frac{5\pi}{6}, 2\pi)$.

6. Using Graphs

Drawing the functions can help you see the solution more clearly. Graphs can show you:

• Where the function meets certain key values.
• The parts above or below the x-axis where the inequality is true.

7. Using Technology

Using software tools or graphing calculators can make this easier. You can plot the functions to visually check the solution sets and points where they intersect, making the whole process faster and simpler.

By using these steps, students can improve their understanding and skills in solving trigonometric inequalities in Year 12, which will help them tackle more advanced math topics in the future.