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54° 240 10 km

54° 240 10 km

Trigonometry (11th Edition)
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.
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A passenger in an airplane at an altitude of 10 kilometers sees two towns directly to the east of the plane. The angles of depression to the towns are 24° and 54° (see figure). How far apart are the towns? (Round your answer to one decimal place.)

### Understanding Angle of Elevation and Depression Using Real-Life Examples #### Diagram Overview: The above diagram demonstrates the concepts of angles of elevation and depression in a real-world scenario using the example of an aircraft and a cityscape. #### Detailed Explanation: - **Aircraft Positioning**: The aircraft is flying at a height of 10 km above the ground level. - **Point of Reference**: There are landmarks on the ground (possibly buildings) used to observe the aircraft. - **Angles Involved**: - **Angle of Depression (54°)**: This is the downward angle from the horizontal line at the aircraft's altitude to a point on the ground. It is the angle formed between the line of sight from the aircraft to the ground and the horizontal line from the aircraft. - **Angle of Elevation (24°)**: This is the upward angle from the ground level to the point of sight of the aircraft. It is formed between the ground observer's line of sight to the aircraft and the horizontal line from the observer. #### Interpretation of the Diagram: 1. **Aircraft Altitude**: - The vertical distance from the aircraft to the ground is consistently shown as 10 km. 2. **Observation Points on the Ground**: - The diagram shows two key observation points beneath the flight path of the aircraft, with respective angles of depression and elevation. 3. **Geometric Relationships**: - **Right Angle Triangle**: The 10 km vertical distance acts as one of the legs of right-angle triangles formed by the angles of depression (54°) and elevation (24°). #### Applications in Real Life: This concept is extensively used in aviation, navigation, and even in determining distances and heights of inaccessible objects. Understanding the angles of elevation and depression can help simplify various complex problems related to geometry and trigonometry. By mastering these principles, students can apply similar methodologies to solve practical problems in physics, engineering, and even urban planning.
Transcribed Image Text:### Understanding Angle of Elevation and Depression Using Real-Life Examples #### Diagram Overview: The above diagram demonstrates the concepts of angles of elevation and depression in a real-world scenario using the example of an aircraft and a cityscape. #### Detailed Explanation: - **Aircraft Positioning**: The aircraft is flying at a height of 10 km above the ground level. - **Point of Reference**: There are landmarks on the ground (possibly buildings) used to observe the aircraft. - **Angles Involved**: - **Angle of Depression (54°)**: This is the downward angle from the horizontal line at the aircraft's altitude to a point on the ground. It is the angle formed between the line of sight from the aircraft to the ground and the horizontal line from the aircraft. - **Angle of Elevation (24°)**: This is the upward angle from the ground level to the point of sight of the aircraft. It is formed between the ground observer's line of sight to the aircraft and the horizontal line from the observer. #### Interpretation of the Diagram: 1. **Aircraft Altitude**: - The vertical distance from the aircraft to the ground is consistently shown as 10 km. 2. **Observation Points on the Ground**: - The diagram shows two key observation points beneath the flight path of the aircraft, with respective angles of depression and elevation. 3. **Geometric Relationships**: - **Right Angle Triangle**: The 10 km vertical distance acts as one of the legs of right-angle triangles formed by the angles of depression (54°) and elevation (24°). #### Applications in Real Life: This concept is extensively used in aviation, navigation, and even in determining distances and heights of inaccessible objects. Understanding the angles of elevation and depression can help simplify various complex problems related to geometry and trigonometry. By mastering these principles, students can apply similar methodologies to solve practical problems in physics, engineering, and even urban planning.
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