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 is the opposite side, the shadow is the adjacent side, and the angle of elevation is the angle between the shadow and the line of sight.
2. Use the tangent function, which relates the opposite side to the adjacent side in a right triangle: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
3. Substitute the known values into the tangent function: \( \tan(\theta) = \frac{34.09}{37.62} \).
4. Solve for \( \theta \) by taking the arctangent (inverse tangent) of both sides: \( \theta = \tan^{-1}\left(\frac{34.09}{37.62}\right) \).
5. Use a calculator to find \( \theta \) to the nearest hundredth of a degree.

Key Concepts

Angle of Elevation

The angle of elevation is the angle formed by the horizontal line from an observer's eye to an object above that line. In this context, it refers to the angle between the ground and the line of sight to the top of the building. Understanding this concept is crucial for applying trigonometric functions to solve problems involving heights and distances.

Trigonometric Ratios

Trigonometric ratios, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. For the problem at hand, the tangent function is particularly relevant, as it relates the angle of elevation to the opposite side (the height of the building) and the adjacent side (the length of the shadow). Mastery of these ratios is essential for solving problems involving right triangles.

Right Triangle Properties

A right triangle has one angle measuring 90 degrees, and the relationships between its sides and angles are governed by trigonometric principles. In this scenario, the building and its shadow form a right triangle, where the height of the building is the opposite side, and the shadow length is the adjacent side. Recognizing and applying the properties of right triangles is fundamental for calculating the angle of elevation.