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Write the sine, cosine and tangent ratios of θ as a decimal correct to four decimal places where required.

To solve this problem, we need to identify the sine, cosine, and tangent ratios for the given angles in each right triangle. These trigonometric ratios are calculated as follows:

- **Sine (sin θ) = Opposite / Hypotenuse**
- **Cosine (cos θ) = Adjacent / Hypotenuse**
- **Tangent (tan θ) = Opposite / Adjacent**

### Triangle I:
- **Opposite side to angle θ**: 3 m
- **Adjacent side to angle θ**: 4 m
- **Hypotenuse**: 5 m

For Triangle I:
1. **sin θ** = Opposite / Hypotenuse = \( \frac{3}{5} = 0.6000 \)
2. **cos θ** = Adjacent / Hypotenuse = \( \frac{4}{5} = 0.8000 \)
3. **tan θ** = Opposite / Adjacent = \( \frac{3}{4} = 0.7500 \)

### Triangle II:
- **Opposite side to angle θ**: 5 m
- **Adjacent side to angle θ**: 12 m
- **Hypotenuse**: 13 m

For Triangle II:
1. **sin θ** = Opposite / Hypotenuse = \( \frac{5}{13} \approx 0.3846 \)
2. **cos θ** = Adjacent / Hypotenuse = \( \frac{12}{13} \approx 0.9231 \)
3. **tan θ** = Opposite / Adjacent = \( \frac{5}{12} \approx 0.4167 \)

These are the sine, cosine, and tangent ratios for each triangle. If you need any further details or have more questions, let me know!

Would you like additional details or have any further questions? Here are some related questions to explore:

1. What is the general formula for finding the hypotenuse using the Pythagorean theorem?
2. How would you solve for the missing side if only the angle θ and one side length were given?
3. Can the sine, cosine, and tangent ratios be used to determine the area of the triangle?
4. What is the significance of these trigonometric ratios in real-world applications?
5. How do you find the angle θ using inverse trigonometric functions?
6. How does the Pythagorean identity relate to the sine and cosine functions?
7. What are the sine, cosine, and tangent values for a 45-degree angle in a right triangle?
8. How do these trigonometric ratios change if the triangle is scaled up or down?

**Tip:** Always ensure your calculator is in the correct mode (degree or radian) when working with trigonometric functions!

Learn how to calculate the sine, cosine, and tangent ratios of angle θ in right triangles with step-by-step solutions and decimal approximations. This problem explores fundamental trigonometric concepts including sine, cosine, and tangent ratios applied to two different triangles.

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