Solve The Trigonometric Inequality Below For The Interval \[$-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\$\]. Use Interval Notation To Represent The Solution Set.\[$\frac{5 \sqrt{3}}{3} \tan (\theta) \geq 5\$\]

Feb 26, 2025 by ADMIN 213 views

**Solving Trigonometric Inequalities: A Step-by-Step Guide**

Introduction

Trigonometric inequalities are a crucial part of mathematics, particularly in trigonometry and calculus. They involve solving equations that contain trigonometric functions, such as sine, cosine, and tangent, and inequalities that involve these functions. In this article, we will focus on solving a specific trigonometric inequality involving the tangent function. We will use the given inequality and solve it for the interval ${-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}}$ . Our goal is to find the solution set in interval notation.

The Given Inequality

The given inequality is:

\frac{5 \sqrt{3}}{3} \tan (\theta) \geq 5

This inequality involves the tangent function, which is a fundamental trigonometric function. The tangent function is defined as the ratio of the sine and cosine functions:

\tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)}

Step 1: Isolate the Tangent Function

To solve the inequality, we need to isolate the tangent function. We can do this by dividing both sides of the inequality by $\frac{5 \sqrt{3}}{3}$ :

\tan (\theta) \geq \frac{5}{\frac{5 \sqrt{3}}{3}}

Simplifying the right-hand side, we get:

\tan (\theta) \geq \frac{3}{\sqrt{3}}

Step 2: Simplify the Right-Hand Side

We can simplify the right-hand side by rationalizing the denominator:

\frac{3}{\sqrt{3}} = \frac{3 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{3 \sqrt{3}}{3} = \sqrt{3}

So, the inequality becomes:

\tan (\theta) \geq \sqrt{3}

Step 3: Find the Solution Set

To find the solution set, we need to find the values of $\theta$ that satisfy the inequality. We can do this by using the inverse tangent function:

\theta \geq \tan^{-1} (\sqrt{3})

The inverse tangent function returns an angle in the range ${-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}}$ . Therefore, the solution set is:

\theta \in \left[ \tan^{-1} (\sqrt{3}), \frac{\pi}{2} \right]

Conclusion

In this article, we solved a trigonometric inequality involving the tangent function. We isolated the tangent function, simplified the right-hand side, and found the solution set in interval notation. The solution set is:

\theta \in \left[ \tan^{-1} (\sqrt{3}), \frac{\pi}{2} \right]

This solution set represents the values of $\theta$ that satisfy the given inequality.

Final Answer

The final answer is $\boxed{\left[ \tan^{-1} (\sqrt{3}), \frac{\pi}{2} \right]}$ .

References

[1] "Trigonometry" by Michael Corral
[2] "Calculus" by Michael Spivak

Additional Resources

Khan Academy: Trigonometry
MIT OpenCourseWare: Calculus
Wolfram Alpha: Trigonometric Inequalities
Frequently Asked Questions (FAQs) on Solving Trigonometric Inequalities ====================================================================

Q: What is a trigonometric inequality?

A: A trigonometric inequality is an equation that contains a trigonometric function, such as sine, cosine, or tangent, and an inequality symbol, such as greater than or equal to.

Q: How do I solve a trigonometric inequality?

A: To solve a trigonometric inequality, you need to isolate the trigonometric function, simplify the right-hand side, and find the solution set in interval notation.

Q: What is the inverse tangent function?

A: The inverse tangent function, denoted as $\tan^{-1} (x)$ , returns an angle in the range ${-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}}$ that satisfies the equation $\tan (\theta) = x$ .

Q: How do I use the inverse tangent function to solve a trigonometric inequality?

A: To use the inverse tangent function to solve a trigonometric inequality, you need to isolate the tangent function, simplify the right-hand side, and then use the inverse tangent function to find the solution set.

Q: What is the solution set in interval notation?

A: The solution set in interval notation is a way of representing the values of the variable that satisfy the inequality. It is typically represented as a closed interval, such as ${a, b}$ , where $a$ and $b$ are the endpoints of the interval.

Q: How do I determine the endpoints of the solution set?

A: To determine the endpoints of the solution set, you need to find the values of the variable that satisfy the inequality. You can do this by using the inverse tangent function, as described earlier.

Q: Can I use a calculator to solve a trigonometric inequality?

A: Yes, you can use a calculator to solve a trigonometric inequality. Many calculators have a built-in inverse tangent function that you can use to find the solution set.

Q: What are some common trigonometric inequalities?

A: Some common trigonometric inequalities include:

$\sin (\theta) \geq \frac{1}{2}$
$\cos (\theta) \leq \frac{1}{2}$
$\tan (\theta) \geq \sqrt{3}$

Q: How do I graph a trigonometric inequality?

A: To graph a trigonometric inequality, you need to graph the corresponding equation and then shade the region that satisfies the inequality.

Q: Can I use a graphing calculator to graph a trigonometric inequality?

A: Yes, you can use a graphing calculator to graph a trigonometric inequality. Many graphing calculators have a built-in graphing function that you can use to graph the inequality.