This worksheet will guide you through solving problems involving angles of elevation and depression using trigonometry. Understanding these concepts is crucial in fields like surveying, navigation, and architecture. We'll cover the basics and work through progressively challenging examples.
What are Angles of Elevation and Depression?
Before we dive into the problems, let's define our key terms:
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Angle of Elevation: This is the angle formed between the horizontal line of sight and the line of sight up to an object. Imagine you're looking up at a bird; the angle from your horizontal gaze to the bird is the angle of elevation.
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Angle of Depression: This is the angle formed between the horizontal line of sight and the line of sight down to an object. Think of looking down from a cliff to a boat; the angle from your horizontal gaze to the boat is the angle of depression.
Important Note: Angles of elevation and depression are always measured from the horizontal. They are also always acute angles (less than 90 degrees).
Key Trigonometric Functions: A Quick Refresher
To solve these problems, we'll primarily use three trigonometric functions:
- Sine (sin): Opposite / Hypotenuse
- Cosine (cos): Adjacent / Hypotenuse
- Tangent (tan): Opposite / Adjacent
Remember, the "opposite" and "adjacent" sides are relative to the angle you're working with in your right-angled triangle.
Example Problems:
Let's tackle some problems to solidify your understanding. Remember to always draw a diagram to visualize the problem!
Problem 1: The Lookout Tower
A park ranger in a lookout tower 20 meters high spots a fire. The angle of depression from the ranger to the fire is 15 degrees. How far is the fire from the base of the tower?
Solution:
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Draw a diagram: Draw a right-angled triangle. The height of the tower (20m) is the opposite side. The distance from the base of the tower to the fire is the adjacent side. The angle of depression is 15 degrees. (Note: The angle of depression from the tower to the fire is equal to the angle of elevation from the fire to the tower.)
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Identify the appropriate trigonometric function: We have the opposite side and need the adjacent side, so we use the tangent function: tan(15°) = opposite / adjacent
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Solve for the adjacent side: Adjacent = opposite / tan(15°) = 20m / tan(15°) ≈ 74.64m
Therefore, the fire is approximately 74.64 meters from the base of the tower.
Problem 2: The Airplane
An airplane is flying at an altitude of 5000 meters. The angle of elevation from an observer on the ground to the airplane is 20 degrees. How far is the airplane from the observer (assuming a straight-line distance)?
Solution:
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Draw a diagram: This time, the altitude is the opposite side, and we need the hypotenuse (the distance from the observer to the airplane).
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Identify the trigonometric function: We use the sine function: sin(20°) = opposite / hypotenuse
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Solve for the hypotenuse: Hypotenuse = opposite / sin(20°) = 5000m / sin(20°) ≈ 14619m
Therefore, the airplane is approximately 14619 meters from the observer.
Practice Problems:
Try these problems to test your skills. Remember to draw a diagram for each one!
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A ladder leans against a wall. The angle of elevation from the ground to the top of the ladder is 70 degrees. The ladder is 10 meters long. How high up the wall does the ladder reach?
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A ship is 3 kilometers from a lighthouse. The angle of elevation from the ship to the top of the lighthouse is 10 degrees. How tall is the lighthouse?
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From the top of a cliff, the angle of depression to a boat is 35 degrees. The cliff is 50 meters high. How far is the boat from the base of the cliff?
This worksheet provides a foundation for understanding and applying trigonometry to angles of elevation and depression. Remember to practice regularly to build your skills and confidence. Good luck!
