This practice exercise focuses on solving problems related to angles of elevation and depression. Remember that the angle of elevation is the angle measured upwards from the horizontal to the line of sight, while the angle of depression is the angle measured downwards from the horizontal to the line of sight. Both angles are always measured with respect to the horizontal.
Understanding the Concepts
Before tackling the problems, let's review the key concepts:
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Right-angled triangles: Most problems involving angles of elevation and depression can be solved using trigonometric ratios (sine, cosine, and tangent) within a right-angled triangle. The angle of elevation or depression is one of the acute angles in the triangle.
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Trigonometric ratios: Recall the definitions:
- sin θ = opposite / hypotenuse
- cos θ = adjacent / hypotenuse
- tan θ = opposite / adjacent
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Identifying the sides: Carefully identify the opposite, adjacent, and hypotenuse sides relative to the given angle in the right-angled triangle. This is crucial for selecting the correct trigonometric ratio.
Practice Problems
Let's work through some examples:
Problem 1: A bird sits on top of a 15-meter tall tree. A person standing 20 meters away from the tree observes the bird. What is the angle of elevation from the person to the bird?
Solution:
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Draw a diagram: Draw a right-angled triangle. The height of the tree (15m) is the opposite side, the distance from the person to the tree (20m) is the adjacent side, and the angle of elevation is the angle we need to find.
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Choose the correct trigonometric ratio: Since we know the opposite and adjacent sides, we use the tangent ratio: tan θ = opposite / adjacent.
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Solve for the angle: tan θ = 15/20. Use a calculator to find the inverse tangent (arctan or tan⁻¹) of (15/20) to find the angle θ.
Problem 2: An airplane is flying at an altitude of 5000 meters. The pilot observes a landmark on the ground at an angle of depression of 15 degrees. What is the horizontal distance between the airplane and the landmark?
Solution:
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Draw a diagram: Draw a right-angled triangle. The altitude of the airplane (5000m) is the opposite side. The horizontal distance is the adjacent side, and the angle of depression is 15 degrees. Note: the angle of depression from the airplane to the landmark is equal to the angle of elevation from the landmark to the airplane.
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Choose the correct trigonometric ratio: We know the opposite side and need to find the adjacent side, so we use the tangent ratio: tan 15° = opposite / adjacent.
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Solve for the adjacent side: Rearrange the equation to solve for the adjacent side (the horizontal distance).
Problem 3: From the top of a lighthouse 75 meters tall, the angle of depression to a boat is 20°. How far is the boat from the base of the lighthouse?
Problem 4: A ladder leans against a wall, making an angle of 70° with the ground. If the foot of the ladder is 2 meters from the wall, how long is the ladder?
Remember: Always draw a diagram to help visualize the problem and accurately identify the sides of the right-angled triangle. Use the correct trigonometric ratio and solve for the unknown value. Make sure your calculator is set to the correct angle mode (degrees or radians).
These practice problems should help you strengthen your understanding of angles of elevation and depression. Remember to practice more problems to become comfortable with the concepts and techniques.
