Draw a sketch showing the given information and determine which trigonometric function can be used to find the distance. Substitute the values in the trigonometric function and find the distance from dock A to the coral reef.

To solve this problem, we need to outline the given information, which likely involves a right triangle and some specific trigonometric function to calculate the distance. Based on your prompt, this could be a typical situation where we are provided with an angle and either one or two sides of a triangle. Here's a guide to approach the problem:

### Step 1: Sketching the Problem
1. **Dock A** and the **coral reef** can be points on a coordinate plane.
2. You can draw a right triangle where:
   - One point (say point \(A\)) represents the **dock**.
   - Another point (say point \(C\)) represents the **coral reef**.
   - The third point represents the perpendicular from the coral reef to the horizontal line (this forms the right angle).
   
   Label the sides accordingly:
   - Let the distance from dock A to the coral reef be the **hypotenuse**.
   - One of the legs could be the horizontal distance from dock A, and the other leg could be a vertical distance (height).
   
   If we know an angle (say angle \(\theta\)) between the dock, coral reef, and the horizontal line, we can use trigonometric ratios.

### Step 2: Choosing the Correct Trigonometric Function
   Based on the given information, select the correct trigonometric function. Trigonometric functions relate angles to side lengths in a right triangle. Here’s a breakdown:
   
   - **Sine (\(\sin\theta\))**: ratio of the opposite side to the hypotenuse.
   - **Cosine (\(\cos\theta\))**: ratio of the adjacent side to the hypotenuse.
   - **Tangent (\(\tan\theta\))**: ratio of the opposite side to the adjacent side.
   
   The choice depends on which sides you know:
   - If you know the angle and the opposite side, use **sine**.
   - If you know the angle and the adjacent side, use **cosine**.
   - If you know both legs, use **tangent**.

### Step 3: Substitute Values into the Trigonometric Function
   Suppose we are given:
   - The angle \(\theta\) between the dock, coral reef, and the horizontal is \(30^\circ\).
   - The adjacent side (horizontal distance) is 100 meters.
   
   To find the distance from dock A to the coral reef (hypotenuse), we would use the **cosine function**:
   
   \[
   \cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}}
   \]
   \[
   \cos(30^\circ) = \frac{100}{\text{distance}}
   \]
   
   Solving for distance:
   \[
   \text{distance} = \frac{100}{\cos(30^\circ)}
   \]
   Using \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\), we get:
   \[
   \text{distance} = \frac{100}{\frac{\sqrt{3}}{2}} = \frac{100 \times 2}{\sqrt{3}} = \frac{200}{\sqrt{3}} \approx 115.47 \text{ meters}
   \]

### Conclusion:
The distance from dock A to the coral reef is approximately **115.47 meters**, assuming the angle is 30° and the adjacent distance is 100 meters.

Would you like to adjust the values, or do you have a specific example in mind? Feel free to share additional details.

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Here are 5 related questions that build upon this:

1. What is the relationship between sine and cosine for complementary angles in a right triangle?
2. How can you use the Pythagorean theorem to verify the calculated distance?
3. What other trigonometric functions can be used if the given information changes?
4. How would you calculate the distance if only the opposite side and angle are known?
5. How can you use inverse trigonometric functions to find the angle if the distance is known?

**Tip**: Always double-check the units of measurement when substituting values to ensure they align correctly (e.g., meters, degrees).

Learn how to find the distance from dock A to a coral reef using trigonometry. This example uses a right triangle and the cosine function, providing a step-by-step solution for calculating the hypotenuse. Suitable for students in Grades 9-11.

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