If the ratio of the length of a pole and its shadow is 1:1 then find the angle of elevation of sun.

Answer

To solve the problem, we need to find the angle of elevation of the sun given that the ratio of the length of a pole to its shadow is 1:1.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a pole (let's denote its height as \( H \)) and its shadow (let's denote the length of the shadow as \( S \)).
- According to the problem, the ratio of the height of the pole to the length of its shadow is \( 1:1 \). This means \( H = S \).

2. **Setting Up the Right Triangle**:
- When the sun is shining, it creates a right triangle where:
- The height of the pole \( H \) is the perpendicular side.
- The length of the shadow \( S \) is the base.
- The angle of elevation of the sun is \( \theta \).

3. **Using the Tangent Function**:
- In a right triangle, the tangent of an angle is defined as the ratio of the opposite side (perpendicular) to the adjacent side (base).
- Therefore, we can write:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{H}{S} \]

4. **Substituting the Values**:
- Since we know that \( H = S \), we substitute:
\[ \tan(\theta) = \frac{H}{H} = 1 \]

5. **Finding the Angle**:
- We know that \( \tan(\theta) = 1 \) corresponds to an angle of \( 45^\circ \).
- Therefore, we conclude that:
\[ \theta = 45^\circ \]

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