Find the angle of elevation of the sun when the shadow of a pole 'h' metres high is `sqrt3h` metres long.

Answer

To find the angle of elevation of the sun when the shadow of a pole \( h \) meters high is \( \sqrt{3}h \) meters long, we can follow these steps:

### Step 1: Draw the Diagram
Draw a right triangle where:
- The vertical side (perpendicular) represents the height of the pole, which is \( h \) meters.
- The horizontal side (base) represents the length of the shadow, which is \( \sqrt{3}h \) meters.
- The angle of elevation of the sun is denoted as \( \theta \).

### Step 2: Identify the Trigonometric Ratio
In the right triangle, we can use the tangent function, which is defined as the ratio of the opposite side (height of the pole) to the adjacent side (length of the shadow):
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{\sqrt{3}h} \]

### Step 3: Simplify the Expression
Now, simplify the expression:
\[ \tan(\theta) = \frac{h}{\sqrt{3}h} = \frac{1}{\sqrt{3}} \]

### Step 4: Find the Angle
To find the angle \( \theta \), we need to determine the angle whose tangent is \( \frac{1}{\sqrt{3}} \). We know from trigonometric values that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
Thus, we can conclude:
\[ \theta = 30^\circ \]

### Step 5: State the Final Answer
The angle of elevation of the sun is \( 30^\circ \).

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