If the height and length of the shadow of a man are the same, then the angle of elevation of the sun is:

Options

A. `45^(@)`
B. `60^(@)`
C. `90^(@)`
D. `120^(@)`

Answer

To solve the problem, we need to find the angle of elevation of the sun when the height of a man and the length of his shadow are equal. Let's break it down step by step.

### Step-by-Step Solution:

1. **Understand the Scenario**:
- Let the height of the man be \( h \).
- Since the height and the length of the shadow are the same, the length of the shadow is also \( h \).

2. **Draw a Right Triangle**:
- When the sun is shining, it creates a right triangle where:
- The height of the man (\( h \)) is the opposite side.
- The length of the shadow (\( h \)) is the adjacent side.
- The angle of elevation of the sun is the angle formed between the line of sight to the top of the man and the horizontal ground.

3. **Use the Tangent Function**:
- The tangent of the angle of elevation \( \theta \) can be expressed as:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{h} \]
- Simplifying this gives:
\[ \tan(\theta) = 1 \]

4. **Find the Angle**:
- We know that \( \tan(45^\circ) = 1 \).
- Therefore, if \( \tan(\theta) = 1 \), then:
\[ \theta = 45^\circ \]

5. **Conclusion**:
- The angle of elevation of the sun is \( 45^\circ \).

### Final Answer:
The angle of elevation of the sun is \( 45^\circ \).

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