What is the angle of elevation of Sun if the ratio of the length of a rod and its shadow is 1:1?

Answer

To solve the problem of finding the angle of elevation of the Sun when the ratio of the length of a rod to its shadow is 1:1, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Let the length of the rod be represented as \( AB \).
- Let the length of the shadow be represented as \( BC \).
- According to the problem, the ratio of the length of the rod to its shadow is \( 1:1 \). This means \( AB = BC \).

2. **Setting Up the Right Triangle**:
- Since the rod is vertical and the shadow is horizontal, we can form a right triangle \( ABC \) where:
- \( AB \) is the height of the rod (perpendicular).
- \( BC \) is the length of the shadow (base).
- \( AC \) is the line from the top of the rod to the tip of the shadow, which represents the line of sight to the Sun.

3. **Using Trigonometric Ratios**:
- In right triangle \( ABC \), we can use the tangent function to find the angle of elevation \( \alpha \):
\[ \tan(\alpha) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC} \]
- Since \( AB = BC \), we can substitute:
\[ \tan(\alpha) = \frac{AB}{AB} = 1 \]

4. **Finding the Angle**:
- We know that \( \tan(45^\circ) = 1 \).
- Therefore, we can conclude that:
\[ \alpha = 45^\circ \]

5. **Final Answer**:
- The angle of elevation of the Sun is \( 45^\circ \).