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The problem asks to find the missing side of the right triangle using the Pythagorean theorem, followed by calculating the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent for angle \( \theta \).

### Step 1: Finding the Missing Side

You are given a right triangle with:
- Opposite side (\( a \)) = 16
- Adjacent side (\( b \)) = 30
- Hypotenuse (\( c \)) = ?

Using the Pythagorean theorem:
\[
c^2 = a^2 + b^2
\]
Substitute the values:
\[
c^2 = 16^2 + 30^2 = 256 + 900 = 1156
\]
\[
c = \sqrt{1156} = 34
\]

So, the hypotenuse (\( c \)) is 34.

### Step 2: Trigonometric Ratios

Now, using the triangle sides, we can find the six trigonometric functions:

- **Sine**: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{16}{34} = \frac{8}{17} \)
- **Cosine**: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{30}{34} = \frac{15}{17} \)
- **Tangent**: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{16}{30} = \frac{8}{15} \)
- **Cosecant**: \( \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{34}{16} = \frac{17}{8} \)
- **Secant**: \( \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{34}{30} = \frac{17}{15} \)
- **Cotangent**: \( \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{30}{16} = \frac{15}{8} \)

### Summary:
- \( \sin(\theta) = \frac{8}{17} \)
- \( \cos(\theta) = \frac{15}{17} \)
- \( \tan(\theta) = \frac{8}{15} \)
- \( \csc(\theta) = \frac{17}{8} \)
- \( \sec(\theta) = \frac{17}{15} \)
- \( \cot(\theta) = \frac{15}{8} \)

Let me know if you would like further details or have any questions!

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Here are 5 related questions to explore:
1. How can you verify the trigonometric values using a calculator?
2. What happens to these ratios if the triangle is scaled up or down?
3. Can you apply these ratios to solve for \( \theta \)?
4. How are trigonometric identities related to these basic ratios?
5. How would the values change if the triangle were not a right triangle?

**Tip**: Always reduce trigonometric ratios to their simplest form for clarity in calculations!

Use the Pythagorean Theorem to find the length of the missing side of the right triangle. Then find the value of each of the six trigonometric functions of θ.

Solve a right triangle using the Pythagorean theorem and find the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) for θ. This problem covers key trigonometry concepts and the Pythagorean theorem, suitable for students in Grades 9-12.

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