A woman standing on a hill sees a flagpole that she knows is...

A woman standing on a hill sees a flagpole that she knows is 60 ft tall. The angle of depression to the bottom of the pole is $14^{\circ},$ and the angle of elevation to the top of the pole is $18^{\circ}.$ Find her distance $x$ from the pole.

Key Concepts

Tangent Function

The tangent function in trigonometry is defined as the ratio of the length of the opposite side to the adjacent side in a right triangle. It is especially useful in problems where one needs to relate an angle with the relative vertical and horizontal distances, such as when calculating the distance from an observer to an object at a different height.

Right Triangle Trigonometry

Right triangle trigonometry involves using the relationships between the angles and sides of a right triangle. In many practical scenarios, such as computing distances and heights, a situation is modeled using right triangles where trigonometric functions become vital tools in solving for unknown quantities.

Angle of Depression

This is the angle formed between a horizontal line from the observer and the line of sight when the observer looks downward. In problems involving heights and distances, the angle of depression is used to set up a right triangle where the observer’s height difference from the target's base can be related to the horizontal distance.

Angle of Elevation

This is the angle formed between a horizontal line from the observer and the line of sight when the observer looks upward. It is used in right triangle trigonometry to relate the observer's sightline to objects above the horizontal, allowing for the calculation of heights and distances.

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