Solve each problem.See Examples 3 and 4. Angle of Elevation of the Sun The length of the shadow of a building 34.09 m tall is 37.62 m. Find the angle of elevation of the sun to the nearest hundredth of a degree.

Verified step by step guidance

1. Identify the right triangle formed by the building, its shadow, and the line of sight to the sun. The building height is the opposite side, the shadow length is the adjacent side, and the angle of elevation is the angle between the ground and the line of sight to the sun.
2. Label the known values: opposite side (height) = 34.09 m, adjacent side (shadow length) = 37.62 m, and the angle of elevation as \( \theta \).
3. Use the tangent function, which relates the opposite side and adjacent side in a right triangle: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). Substitute the known values: \( \tan(\theta) = \frac{34.09}{37.62} \).
4. To find the angle \( \theta \), take the inverse tangent (arctangent) of the ratio: \( \theta = \tan^{-1} \left( \frac{34.09}{37.62} \right) \).
5. Use a calculator to evaluate the inverse tangent and round the result to the nearest hundredth of a degree to find the angle of elevation.

Key Concepts

Angle of Elevation

The angle of elevation is the angle formed between the horizontal ground and the line of sight to an object above the horizontal. In this problem, it is the angle between the ground and the sun’s rays as they hit the top of the building.

Right Triangle Trigonometry

The building, its shadow, and the line from the top of the building to the tip of the shadow form a right triangle. Understanding how to apply trigonometric ratios like tangent, which relates opposite and adjacent sides, is essential to find the angle.

Tangent Function

The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side. Here, tangent(angle) = height of building / length of shadow. Using the inverse tangent function allows calculation of the angle of elevation.