Pat needs to determine the height of a tree before cutting...

Pat needs to determine the height of a tree before cutting it down to be sure that it will not fall on a nearby fence. The angle of elevation of the tree from one position on a flat path from the tree is $30^{\circ}$, and from a second position 40 feet farther along this path it is $20^{\circ} .$ What is the height of the tree?

Key Concepts

Tangent Function

The tangent function relates the angle in a right triangle to the ratio of the opposite side (often representing height in these problems) to the adjacent side (representing horizontal distance). This trigonometric ratio is particularly useful for solving problems where the height of an object is to be determined using the measured angle of elevation.

Systems of Equations in Trigonometry

When measurements are taken from different points which yield different angles of elevation to the same object, each observation forms a separate trigonometric equation. These equations can be combined into a system of equations, which, when solved, allow for the determination of unknown quantities such as the height or distance of an object.

Right Triangle Trigonometry

This concept involves using the relationships between the angles and sides of right triangles to solve for unknown measurements. It is fundamental in many real-world applications where indirect measurement is needed—such as determining the height of an object—by connecting known distances and angles using trigonometric ratios.

Angle of Elevation

The angle of elevation is the angle formed between a horizontal line of sight and the line drawn from the observer's eye to an object above the horizontal line. It is a key measurement in problems dealing with heights and distances that leverages the principles of right triangle trigonometry.

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