Understanding the Angle of Elevation Problem

This problem asks us to find the angle at which the sun's rays hit the ground, creating a shadow of a known length from a pole of a known height. This situation forms a right-angled triangle, where:

• The height of the pole is one leg (opposite to the angle of elevation).
• The length of the shadow is the other leg (adjacent to the angle of elevation).
• The angle of elevation is the acute angle formed at the base of the pole (where the shadow ends) between the ground and the line of sight to the top of the pole.

Setting up the Trigonometric Relationship

In a right-angled triangle, the relationship between the opposite side, the adjacent side, and the angle is given by the tangent function. The formula is:

\(\tan(\theta) = \frac{\text{Opposite Side}}{\text{Adjacent Side}}\)

Where:

• \(\theta\) is the angle of elevation we need to find.
• The opposite side is the height of the pole, which is 9 m.
• The adjacent side is the length of the shadow, which is \(3\sqrt 3 \) m.

Calculating the Angle of Elevation

Now, let's substitute the given values into the tangent formula:

\(\tan(\theta) = \frac{9 \text{ m}}{3\sqrt 3 \text{ m}}\)

We can simplify this expression:

\(\tan(\theta) = \frac{9}{3\sqrt 3}\)

First, divide the numerator and the denominator by 3:

\(\tan(\theta) = \frac{3}{\sqrt 3}\)

To make the denominator a rational number, we can multiply both the numerator and the denominator by \(\sqrt 3\):

\(\tan(\theta) = \frac{3}{\sqrt 3} \times \frac{\sqrt 3}{\sqrt 3}\)

\(\tan(\theta) = \frac{3\sqrt 3}{3}\)

Now, cancel out the 3 from the numerator and the denominator:

\(\tan(\theta) = \sqrt 3\)

We need to find the angle \(\theta\) whose tangent is \(\sqrt 3\). We recall the standard trigonometric values for common angles. The tangent of 60° is \(\sqrt 3\).

So,

\(\theta = \tan^{-1}(\sqrt 3)\)

\(\theta = 60^\circ\)

Therefore, the angle of elevation of the sun is 60°.

Summary of Steps

Here's a quick overview of the steps taken to find the angle of elevation:

1. Identify the height of the pole as the opposite side and the shadow length as the adjacent side in a right-angled triangle.
2. Recognize that the tangent function relates the opposite and adjacent sides to the angle of elevation.
3. Set up the equation \(\tan(\theta) = \frac{\text{Height of Pole}}{\text{Length of Shadow}}\).
4. Substitute the given values and simplify the expression for \(\tan(\theta)\).
5. Determine the angle \(\theta\) whose tangent is equal to the calculated value (\(\sqrt 3\)).

Revision Table: Key Trigonometric Ratios

Additional Information: Angle of Elevation and Depression

The angle of elevation is the angle measured upwards from the horizontal line of sight to an object above the observer. In contrast, the angle of depression is the angle measured downwards from the horizontal line of sight to an object below the observer.

Both angles are crucial in trigonometry and are used extensively in surveying, navigation, physics, and engineering to solve problems involving heights, distances, and angles.

When dealing with problems like the height of a building or pole and its shadow, the angle of elevation of the sun is typically the angle formed at the ground level.