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.