https://file.aiquickdraw.com/mathbot/file/17267593745246lhrigut.jpg

The problem asks to find the equation for \(\tan(\theta)\) given that \(\sec(\theta) = \frac{5}{3}\) and that the angle \(\theta\) is in Quadrant I. Let's go step by step:

1. **Secant and Cosine Relation:**
   \[
   \sec(\theta) = \frac{1}{\cos(\theta)}
   \]
   Given that \(\sec(\theta) = \frac{5}{3}\), we can express \(\cos(\theta)\) as:
   \[
   \cos(\theta) = \frac{3}{5}
   \]

2. **Pythagorean Identity:**
   The Pythagorean identity involving tangent and secant is:
   \[
   1 + \tan^2(\theta) = \sec^2(\theta)
   \]
   Since \(\sec(\theta) = \frac{5}{3}\), squaring this gives:
   \[
   \sec^2(\theta) = \left(\frac{5}{3}\right)^2 = \frac{25}{9}
   \]

3. **Solving for \(\tan^2(\theta)\):**
   Plug this into the Pythagorean identity:
   \[
   1 + \tan^2(\theta) = \frac{25}{9}
   \]
   Subtract 1 (or \(\frac{9}{9}\)) from both sides:
   \[
   \tan^2(\theta) = \frac{25}{9} - \frac{9}{9} = \frac{16}{9}
   \]

4. **Conclusion:**
   The equation that matches the Pythagorean identity is:
   \[
   1 + \tan^2(\theta) = \left(\frac{5}{3}\right)^2
   \]
   Therefore, the correct answer is the last option:
   \[
   1 + \tan^2(\theta) = \left(\frac{5}{3}\right)^2
   \]

Let me know if you need more details or further clarifications!

---

Here are 5 related questions:
1. How is \(\sec(\theta)\) related to \(\cos(\theta)\)?
2. What are the key trigonometric identities used to solve problems involving \(\sec(\theta)\)?
3. How do you calculate \(\tan(\theta)\) if \(\sec(\theta)\) is given?
4. Why is the Pythagorean identity important in trigonometry?
5. Can you derive \(\tan(\theta)\) directly from \(\sec(\theta)\)?

**Tip:** Always remember the basic Pythagorean identities involving sine, cosine, and tangent, as they are essential in solving trigonometric equations!

The angle θ is in Quadrant I with sec(θ) = 5/3. Which equation could be used to find tan(θ)?

Explore how to find the equation for tan(θ) given sec(θ) = 5/3 in Quadrant I using trigonometric identities. Learn key formulas, including the Pythagorean identity, in a detailed step-by-step explanation. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation