tg(x) less or equal to of sqrt(3) in the geometric circle

To solve the inequality \(\tan(x) \leq \sqrt{3}\) on the unit circle, let's follow these steps:

1. **Recall the unit circle properties**: The tangent of an angle \(x\) in the unit circle is defined as the ratio \(\frac{\sin(x)}{\cos(x)}\). Since \(\tan(x) = \frac{\sin(x)}{\cos(x)}\), the values of \(\tan(x)\) can be positive or negative depending on the quadrant.

2. **Identify where \(\tan(x) = \sqrt{3}\)**: Tangent achieves the value \(\sqrt{3}\) at specific angles. From trigonometric tables or by knowing standard angles:
   \[
   \tan\left(\frac{\pi}{3}\right) = \sqrt{3}
   \]
   This occurs in the first quadrant.

3. **Solve the inequality**: You are asked to find where \(\tan(x) \leq \sqrt{3}\), which includes both where \(\tan(x) = \sqrt{3}\) and where \(\tan(x)\) is less than \(\sqrt{3}\).

   - In the **first quadrant** (from \(0\) to \(\frac{\pi}{2}\)): \(\tan(x)\) starts from 0 at \(x = 0\) and increases, reaching \(\sqrt{3}\) at \(x = \frac{\pi}{3}\).
   - In the **second quadrant** (from \(\frac{\pi}{2}\) to \(\pi\)): \(\tan(x)\) is negative.
   - In the **third quadrant** (from \(\pi\) to \(\frac{3\pi}{2}\)): \(\tan(x)\) is negative.
   - In the **fourth quadrant** (from \(\frac{3\pi}{2}\) to \(2\pi\)): \(\tan(x)\) approaches 0 from below.

4. **Conclusion**: The solution to the inequality \(\tan(x) \leq \sqrt{3}\) on the unit circle is:
   \[
   x \in \left[0, \frac{\pi}{3}\right] \cup \left[\pi, 2\pi\right)
   \]
   This means the inequality is satisfied in the first quadrant from 0 to \(\frac{\pi}{3}\) and the entire third and fourth quadrants.

Would you like more details or have any questions about this? 

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**Related Questions:**
1. How does the tangent function behave in each quadrant?
2. What are the key properties of the unit circle?
3. How do we derive the exact values of \(\tan(x)\) for standard angles?
4. What is the domain and range of the tangent function?
5. How do you solve inequalities involving other trigonometric functions?

**Tip**: When working with trigonometric functions, it's often helpful to visualize them on the unit circle to understand their behavior in different quadrants.

tg(x) less or equal to sqrt(3) in the geometric circle

Explore the solution to the trigonometric inequality \(\tan(x) \leq \sqrt{3}\) on the unit circle. This step-by-step guide covers key concepts like the tangent function, its behavior in different quadrants, and how to solve inequalities involving trigonometric functions. Suitable for students in Grades 10-12.

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