https://file.aiquickdraw.com/mathbot/file/17303887057631jezx7xl.jpg

To solve for the trigonometric ratios given in the image, we’ll use the right triangle properties. Let’s identify the lengths and calculate each trigonometric function.

### Step 1: Find the Hypotenuse
Given:
- Opposite side to \( \alpha \) (adjacent to \( \beta \)) = 8
- Adjacent side to \( \alpha \) (opposite to \( \beta \)) = 14

Using the Pythagorean theorem to find the hypotenuse \( c \):
\[
c = \sqrt{8^2 + 14^2} = \sqrt{64 + 196} = \sqrt{260} = 2\sqrt{65} \approx 16.12
\]

### Step 2: Calculate the Trigonometric Ratios

1. **\( \sin(\alpha) \) and \( \cos(\beta) \):**
   - \( \sin(\alpha) = \frac{\text{opposite to } \alpha}{\text{hypotenuse}} = \frac{8}{16.12} \approx 0.496 \)
   - \( \cos(\beta) = \sin(\alpha) \), so \( \cos(\beta) \approx 0.496 \)

2. **\( \tan(\alpha) \) and \( \cot(\beta) \):**
   - \( \tan(\alpha) = \frac{\text{opposite to } \alpha}{\text{adjacent to } \alpha} = \frac{8}{14} = \frac{4}{7} \approx 0.571 \)
   - \( \cot(\beta) = \tan(\alpha) \), so \( \cot(\beta) \approx 0.571 \)

3. **\( \sec(\alpha) \) and \( \csc(\beta) \):**
   - \( \sec(\alpha) = \frac{\text{hypotenuse}}{\text{adjacent to } \alpha} = \frac{16.12}{14} \approx 1.15 \)
   - \( \csc(\beta) = \sec(\alpha) \), so \( \csc(\beta) \approx 1.15 \)

### Final Answers

(a) \( \sin(\alpha) \approx 0.496 \) and \( \cos(\beta) \approx 0.496 \)  
(b) \( \tan(\alpha) \approx 0.571 \) and \( \cot(\beta) \approx 0.571 \)  
(c) \( \sec(\alpha) \approx 1.15 \) and \( \csc(\beta) \approx 1.15 \)  

Would you like a more detailed explanation or have any questions?

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Here are five related questions to expand your understanding:

1. What is the Pythagorean identity, and how is it applied in trigonometry?
2. How are trigonometric functions related for complementary angles in a right triangle?
3. What are the exact values for trigonometric functions at common angles (like 30°, 45°, and 60°)?
4. How can we use the unit circle to find trigonometric values of angles beyond 90°?
5. What are the reciprocal trigonometric functions, and how are they derived?

**Tip:** Memorizing the basic trigonometric ratios for special triangles, like 45-45-90 and 30-60-90, can speed up your calculations.

Find sin(α) and cos(β), tan(α) and cot(β), and sec(α) and csc(β) for the given right triangle with legs 8 and 14.

Learn how to find sin(α), cos(β), tan(α), cot(β), sec(α), and csc(β) for a right triangle with legs measuring 8 and 14. This problem covers trigonometric ratios and the Pythagorean theorem, suitable for students in Grades 9-11.

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