https://file.aiquickdraw.com/mathbot/file/1729920832215flup5ijr.jpg

To solve this problem, we need to use trigonometric identities, particularly the double-angle formulas:

1. **Given:**
   \[
   \sin(x) = \frac{2}{7}
   \]
   with \( x \) in Quadrant I.

2. **Find \( \cos(x) \):**
   Since \( x \) is in Quadrant I, both \( \sin(x) \) and \( \cos(x) \) are positive. Using the Pythagorean identity:
   \[
   \cos^2(x) = 1 - \sin^2(x)
   \]
   \[
   \cos^2(x) = 1 - \left(\frac{2}{7}\right)^2 = 1 - \frac{4}{49} = \frac{45}{49}
   \]
   Therefore:
   \[
   \cos(x) = \sqrt{\frac{45}{49}} = \frac{\sqrt{45}}{7} = \frac{3\sqrt{5}}{7}
   \]

3. **Compute \( \sin(2x) \):**
   Use the double-angle formula for sine:
   \[
   \sin(2x) = 2 \sin(x) \cos(x)
   \]
   Substituting the values:
   \[
   \sin(2x) = 2 \cdot \frac{2}{7} \cdot \frac{3\sqrt{5}}{7} = \frac{12\sqrt{5}}{49}
   \]

4. **Compute \( \cos(2x) \):**
   Use the double-angle formula for cosine:
   \[
   \cos(2x) = \cos^2(x) - \sin^2(x)
   \]
   Substitute the values:
   \[
   \cos(2x) = \frac{45}{49} - \frac{4}{49} = \frac{41}{49}
   \]

5. **Compute \( \tan(2x) \):**
   Use the identity \( \tan(2x) = \frac{\sin(2x)}{\cos(2x)} \):
   \[
   \tan(2x) = \frac{\frac{12\sqrt{5}}{49}}{\frac{41}{49}} = \frac{12\sqrt{5}}{41}
   \]

### Final Answers:
- \( \sin(2x) = \frac{12\sqrt{5}}{49} \)
- \( \cos(2x) = \frac{41}{49} \)
- \( \tan(2x) = \frac{12\sqrt{5}}{41} \)

Would you like more details on any of these steps?

---

Here are 5 related questions to deepen your understanding:
1. How would these calculations change if \( x \) were in a different quadrant?
2. What would happen if \( \sin(x) \) was given as a fraction involving a square root?
3. How can you derive double-angle identities from basic trigonometric identities?
4. Why do we only consider the positive root for \( \cos(x) \) here?
5. How does the value of \( \tan(2x) \) relate to the values of \( \tan(x) \)?

**Tip:** Memorizing the Pythagorean and double-angle identities can significantly speed up solving similar trigonometric problems.

If \( \sin x = \frac{2}{7} \), x in quadrant I, then find (without finding x) \( \sin(2x) \), \( \cos(2x) \), and \( \tan(2x) \).

Learn how to find \( \sin(2x) \), \( \cos(2x) \), and \( \tan(2x) \) given that \( \sin(x) = \frac{2}{7} \) in Quadrant I, using double-angle identities and the Pythagorean identity. Suitable for high school trigonometry students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation