If the horizontal distance of the boat from the bridge is 25 m and the height of the bridge is 25 m, then find the angle of depression of the boat from the bridge.

Answer

To find the angle of depression of the boat from the bridge, we can follow these steps:

### Step 1: Understand the scenario
We have a bridge that is 25 meters high, and a boat that is 25 meters away horizontally from the base of the bridge. We need to find the angle of depression from the top of the bridge to the boat.

### Step 2: Draw a diagram
Draw a right triangle where:
- The vertical side represents the height of the bridge (25 m).
- The horizontal side represents the distance from the base of the bridge to the boat (25 m).
- The angle of depression is the angle formed between the horizontal line from the bridge and the line of sight to the boat.

### Step 3: Identify the right triangle
In the right triangle formed:
- The opposite side (height of the bridge) = 25 m
- The adjacent side (horizontal distance to the boat) = 25 m

### Step 4: Use trigonometric ratios
To find the angle of depression (let's denote it as θ), we can use the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{25}{25} \]

### Step 5: Calculate the tangent
\[ \tan(\theta) = 1 \]

### Step 6: Find the angle
To find θ, we take the arctangent (inverse tangent) of 1:
\[ \theta = \tan^{-1}(1) \]
From trigonometric tables or a calculator, we know:
\[ \theta = 45^\circ \]

### Conclusion
The angle of depression from the bridge to the boat is \( 45^\circ \).

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