https://file.aiquickdraw.com/mathbot/file/17285913322293pq6vhsn.jpg

We are given that \( \tan \theta = \frac{5}{3} \) and \( \cos \theta > 0 \). This information will help us solve for \( \sec \theta \) and \( \sin \theta \).

### Step 1: Identify the correct quadrant
Since \( \tan \theta = \frac{5}{3} \) and \( \cos \theta > 0 \), the angle \( \theta \) must lie in the **first quadrant**, where both sine and cosine are positive.

### Step 2: Use the definition of tangent
We know that:
\[
\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{3}
\]
This means that the opposite side of the right triangle is 5, and the adjacent side is 3.

### Step 3: Find the hypotenuse using the Pythagorean theorem
The hypotenuse \( r \) can be found using:
\[
r = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34}
\]

### Step 4: Calculate \( \sec \theta \) and \( \sin \theta \)
- \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{3}{\sqrt{34}} \)
- \( \sec \theta = \frac{1}{\cos \theta} = \frac{\sqrt{34}}{3} \)
- \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{\sqrt{34}} \)

### Final answers:
- \( \sec \theta = \frac{\sqrt{34}}{3} \)
- \( \sin \theta = \frac{5}{\sqrt{34}} \)

Would you like more details on any step, or do you have any questions?

Here are 5 follow-up questions:
1. How do you find the other trigonometric ratios based on the information provided?
2. What does it mean geometrically for \( \tan \theta = \frac{5}{3} \)?
3. Can you explain why \( \cos \theta \) must be positive in this case?
4. How can the Pythagorean identity be used to verify your trigonometric values?
5. What changes in the problem if \( \cos \theta < 0 \)?

**Tip**: Always check the quadrant when working with trigonometric functions to ensure you have the correct sign for your values.

Let θ be an angle such that tan(θ) = 5/3 and cos(θ) > 0. Find the exact values of sec(θ) and sin(θ).

Learn how to find the exact values of sec(θ) and sin(θ) given that tan(θ) = 5/3 and cos(θ) is positive. This problem uses trigonometric definitions and the Pythagorean Theorem. 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