Apply trigonometric functions to solve real-world problems in navigation, engineering, and scientific calculations using angles of elevation and depression.
Key concepts for solving real-world trigonometry problems
Definition:
The angle between the horizontal line and the line of sight when looking upward at an object.
Example:
Looking up at the top of a building from the ground. The angle between your line of sight and the ground is the angle of elevation.
Key Point:
Always measured from the horizontal line upward.
Definition:
The angle between the horizontal line and the line of sight when looking downward at an object.
Example:
Looking down from the top of a building at a car on the street. The angle between your line of sight and the horizontal is the angle of depression.
Key Point:
Always measured from the horizontal line downward.
Using trigonometry to find heights of tall objects
Scenario:
A person stands 50 meters from a building and measures the angle of elevation to the top as 30°.
Given:
• Distance from building: 50m
• Angle of elevation: 30°
• Person's eye height: 1.5m
Solution:
• Use tan θ = opposite / adjacent
• tan 30° = height / 50
• 0.577 = height / 50
• height = 50 × 0.577 = 28.85m
• Total building height = 28.85 + 1.5 = 30.35m
Scenario:
A surveyor measures the angle of elevation to the top of a tree as 45° from a point 20 meters away.
Given:
• Distance from tree: 20m
• Angle of elevation: 45°
• Surveyor's eye height: 1.6m
Solution:
• Use tan θ = opposite / adjacent
• tan 45° = height / 20
• 1 = height / 20
• height = 20m
• Total tree height = 20 + 1.6 = 21.6m
Finding distances using trigonometric functions
Scenario:
A lighthouse is 100 meters tall. A boat measures the angle of elevation to the top of the lighthouse as 15°.
Question:
How far is the boat from the base of the lighthouse?
Solution:
• Use tan θ = opposite / adjacent
• tan 15° = 100 / distance
• 0.268 = 100 / distance
• distance = 100 / 0.268 = 373.1m
Scenario:
An airplane is flying at an altitude of 3000 meters. The angle of depression to a point on the ground is 25°.
Question:
How far is the airplane from the point on the ground?
Solution:
• Use tan θ = opposite / adjacent
• tan 25° = 3000 / distance
• 0.466 = 3000 / distance
• distance = 3000 / 0.466 = 6433.5m
How trigonometry is used in navigation and GPS
How it works:
GPS satellites use trigonometry to calculate your position on Earth by measuring distances and angles.
Example:
If you're 5 km from point A and 7 km from point B, and the angle between them is 60°, trigonometry helps determine your exact location.
How it works:
Sailors use trigonometry to navigate by measuring angles to stars, lighthouses, and landmarks.
Example:
A ship measures the angle of elevation to a lighthouse as 20°. If the lighthouse is 50m tall, the ship can calculate its distance from shore.
Apply trigonometric concepts to solve real-world problems
Given: A person stands 40 meters from a tower and measures the angle of elevation to the top as 60°.
Find the height of the tower (ignore the person's height).
Solution:
• Use tan θ = opposite / adjacent
• tan 60° = height / 40
• √3 = height / 40
• height = 40 × √3 = 40 × 1.732 = 69.28m
Answer: The tower is 69.28 meters tall.
Given: A lighthouse is 80 meters tall. A ship measures the angle of elevation to the top as 30°.
How far is the ship from the lighthouse?
Solution:
• Use tan θ = opposite / adjacent
• tan 30° = 80 / distance
• 1/√3 = 80 / distance
• distance = 80 × √3 = 80 × 1.732 = 138.56m
Answer: The ship is 138.56 meters from the lighthouse.
Congratulations! You've mastered similarity and trigonometry fundamentals