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In this section, we explore how trigonometry helps us determine heights and distances using angles of elevation and depression. Key concepts include understanding lines of sight, and utilizing trigonometric ratios like tan, sin, and cos to solve real-world problems involving various objects and angles.
In this section, we explore some practical applications of trigonometry, particularly in measuring heights and distances. We begin by revisiting key concepts such as:
We distinguish between angles of elevation and depression through visual aids and diagrams. To calculate heights without direct measurement, we use trigonometric ratios based on known distances and angles. For example, in determining the height of a tower or building using the known distance from the observer and the angle of elevation to the top of that structure. We solve several illustrative examples, demonstrating how to apply trigonometry in practical scenarios, and a series of exercises for reinforcement. Overall, heights and distances are critical applications of trigonometric principles that facilitate measuring objects in various real-life situations.
Line of Sight: The direct line from an observer to the object being viewed.
Angle of Elevation: Angle from the horizontal line up to the object.
Angle of Depression: Angle from the horizontal line down to the object.
Trigonometric Ratios: Ratios used in triangles to calculate lengths and angles.
To find the height you need to see, look up and angle, that’s the key.
Once there was a boy who wanted to measure a tower's height. He learned to look up, using angles bright!
E for Elevation, looking Up; D for Depression, looking Down.
If a tower 15 m away has an angle of elevation of 60°, use tan(60°) = opposite/adjacent to find the height.
Using angles of depression, if the observer's height is known, we can calculate distances to objects below.
Term: Line of Sight
Definition: The line drawn from the eye of an observer to the point in the object viewed by the observer.
The line drawn from the eye of an observer to the point in the object viewed by the observer.
Term: Angle of Elevation
Definition: The angle formed by the line of sight with the horizontal when the object being viewed is above the horizontal level.
The angle formed by the line of sight with the horizontal when the object being viewed is above the horizontal level.
Term: Angle of Depression
Definition: The angle formed by the line of sight with the horizontal when the object being viewed is below the horizontal level.
The angle formed by the line of sight with the horizontal when the object being viewed is below the horizontal level.
Term: Trigonometric Ratios
Definition: Ratios of the lengths of the sides of a right triangle, commonly used to find unknown side lengths or angles.
Ratios of the lengths of the sides of a right triangle, commonly used to find unknown side lengths or angles.