If the height of a vertical pole is $\sqrt{3}$ times the length of its shadow on the ground, then the angle of elevation of the sun at that time is :

Question

Accepted Answer

To determine the angle of elevation of the sun when the height of a vertical pole is $\sqrt{3}$ times the length of its shadow, we can use trigonometric principles.

### Explanation for Option 1: $60^\circ$

1. **Understanding the Problem:**
   - Let the height of the pole be $h$.
   - Let the length of the shadow be $s$.
   - According to the problem, $h = \sqrt{3} \times s$.

2. **Trigonometric Relationship:**
   - The angle of elevation $\theta$ is the angle formed between the line of sight from the tip of the shadow to the top of the pole and the horizontal ground.
   - In a right triangle formed by the pole, the shadow, and the line of sight, the tangent of the angle of elevation is given by:
     $$
     \tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{h}{s}
     $$
   - Substituting the given relationship $h = \sqrt{3} \times s$, we have:
     $$
     \tan(\theta) = \frac{\sqrt{3} \times s}{s} = \sqrt{3}
     $$

3. **Determining the Angle:**
   - The angle $\theta$ for which $\tan(\theta) = \sqrt{3}$ is a well-known trigonometric value.
   - $\tan(60^\circ) = \sqrt{3}$.
   - Therefore, the angle of elevation $\theta$ is $60^\circ$.

### Explanation for Why Other Options are Incorrect:

- **Option 2: $30^\circ$**
  - $\tan(30^\circ) = \frac{1}{\sqrt{3}}$, not $\sqrt{3}$.
  - This does not satisfy the condition $\tan(\theta) = \sqrt{3}$.

- **Option 3: $75^\circ$**
  - $\tan(75^\circ)$ is not equal to $\sqrt{3}$.
  - The exact value of $\tan(75^\circ)$ is $\frac{2 + \sqrt{3}}{1 - \sqrt{3}}$, which is not $\sqrt{3}$.

- **Option 4: $45^\circ$**
  - $\tan(45^\circ) = 1$, not $\sqrt{3}$.
  - This does not match the given condition.

In conclusion, the correct angle of elevation of the sun, given that the height of the pole is $\sqrt{3}$ times the length of its shadow, is $60^\circ$. Therefore, Option 1 is the correct answer.