The height of a tree is `sqrt(3)` times the length of its shadow. Find the angle of elevation of the sun

Answer

To solve the problem, we need to find the angle of elevation of the sun given that the height of the tree is \(\sqrt{3}\) times the length of its shadow. Let's break this down step by step.

### Step 1: Define the Variables
Let:
- \( h \) = height of the tree
- \( s \) = length of the shadow

According to the problem, we have:
\[ h = \sqrt{3} \times s \]

### Step 2: Set Up the Right Triangle
In the right triangle formed by the tree and its shadow:
- The height of the tree \( h \) is the opposite side.
- The length of the shadow \( s \) is the adjacent side.
- The angle of elevation of the sun is \( \theta \).

### Step 3: Use the Tangent Function
Using the definition of the tangent function in a right triangle:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{s} \]

### Step 4: Substitute the Value of \( h \)
Substituting the expression for \( h \) into the tangent function:
\[ \tan(\theta) = \frac{\sqrt{3} \cdot s}{s} \]

### Step 5: Simplify the Expression
The \( s \) in the numerator and denominator cancels out:
\[ \tan(\theta) = \sqrt{3} \]

### Step 6: Find the Angle \( \theta \)
We know from trigonometric values that:
\[ \tan(60^\circ) = \sqrt{3} \]
Thus, we can conclude that:
\[ \theta = 60^\circ \]

### Final Answer
The angle of elevation of the sun is \( 60^\circ \).

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