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To find \(\sec \theta\) and \(\csc \theta\) for the given right triangle, we can proceed as follows:

1. The given triangle has:
   - Opposite side to \(\theta\): 6
   - Hypotenuse: 10
   - Adjacent side to \(\theta\): not given but can be calculated if needed.

2. To find \(\sec \theta\) and \(\csc \theta\):
   - \(\sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent}}\)
   - \(\csc \theta = \frac{\text{Hypotenuse}}{\text{Opposite}}\)

3. For \(\csc \theta\):
   \[
   \csc \theta = \frac{10}{6} = \frac{5}{3}
   \]

4. To find the adjacent side (since we need it for \(\sec \theta\)), use the Pythagorean theorem:
   \[
   \text{Adjacent}^2 + 6^2 = 10^2 \Rightarrow \text{Adjacent}^2 + 36 = 100 \Rightarrow \text{Adjacent} = 8
   \]

5. For \(\sec \theta\):
   \[
   \sec \theta = \frac{10}{8} = \frac{5}{4}
   \]

So, the answers are:
- \(\sec \theta = \frac{5}{4}\)
- \(\csc \theta = \frac{5}{3}\)

Would you like further details or have any questions?

Here are some related questions for further exploration:
1. How would you find \(\tan \theta\) and \(\cot \theta\) for the same triangle?
2. Can you verify the values by calculating the approximate angle \(\theta\) itself?
3. What would \(\sec \theta\) and \(\csc \theta\) be if the triangle's hypotenuse were different?
4. How do these ratios change if \(\theta\) is acute versus obtuse in other contexts?
5. How can you use these trigonometric identities in real-world applications?

**Tip:** Always check if the sides satisfy the Pythagorean theorem to confirm it's a right triangle.

Find \(\sec \theta\) and \(\csc \theta\) for the given right triangle with sides of 6, 10, and an unknown adjacent side.

Learn how to find sec(θ) and csc(θ) for a right triangle with given side lengths. This solution uses trigonometric ratios and the Pythagorean Theorem, ideal for high school students in Grades 9-10.

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