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Interactive Audio Lesson
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Introduction to Angles of Depression
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Today we'll discuss angles of depression and how they apply to calculating heights in surveying. Who can tell me what an angle of depression is?
Is it the angle formed from the horizontal line down to an object below?
Exactly! It measures the angle from your line of sight straight out, down to an object below. It's crucial for height calculations. Now, if we see two ships, one at each angle, what do we need to find the height of the lighthouse?
We need the distance between the ships and the angles of depression to them.
Right! With the correct distance and those angles, we can derive the height of the lighthouse using trigonometric functions.
What formula do we use for that?
We use the formula: Height = Distance / (cot(angle 1) - cot(angle 2)). Let’s work through the example together.
Solving the Problem
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Here’s the data: the distance between the two ships is 100 m. The first angle is 30 degrees and the second is 45 degrees. Let's calculate the height now.
So first we find cotangent for both angles?
That's correct. Remember, cot(30°) is equal to √3 and cot(45°) is 1. What do we get?
Cot(30°) - Cot(45°) = √3 - 1.
Exactly! Now, substituting back into the height formula, what do we have?
Height = 100 / (√3 - 1)!
That's it! Now, calculate the numerical value of the height.
The height is 50 m!
Well done! So, in summary, we used the angles of depression and the distance to find the height of the lighthouse, which was 50 m.
Introduction & Overview
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Quick Overview
Standard
In this section, the height of a lighthouse is calculated by analyzing the angles of depression to two ships located a known distance apart. The trigonometric method applied illustrates how to derive the height from angle measurements.
Detailed
Height Calculation of a Lighthouse
In the example presented here, the problem revolves around finding the height of a lighthouse based on the observed angles of depression to two ships that are 100 meters apart. The angles of depression are measured at 30 degrees and 45 degrees, respectively. By utilizing basic trigonometric principles and the cotangent function, the height can be derived using the formula:
Height = Distance / (cot(original angle) - cot(final angle))
Plugging in the values:
- Distance = 100 m
- Angle 1 (30 degrees) and Angle 2 (45 degrees)
The solution yields a lighthouse height of 50 m, which illustrates a practical application of trigonometry in surveying and engineering contexts.
Audio Book
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Introduction to the Problem
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From the top of a lighthouse, the angles of depression of two ships are 30° and 45°. The two ships, as it was observed from the top of the lighthouse, were 100 m apart. Find the height of the lighthouse.
Detailed Explanation
In this example, we need to determine the height of a lighthouse based on the angles of depression to two ships located at different angles. The angle of depression is the angle formed by a horizontal line from the observer's eye to the object being observed below the horizon. The two angles given are 30° and 45°, indicating how steeply the line of sight drops down from the top of the lighthouse to the ships. Furthermore, we know the distance between the ships is 100 meters.
Examples & Analogies
Imagine you're standing on the edge of a cliff looking down at two boats in the sea. The first boat is at a 30° angle away from you and the second at a steeper 45°. The cliff's height represents your view point, and you want to accurately calculate how tall the cliff is using the angles and distance to the boats.
Applying the Formula to Determine Height
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Height = Distance / [cot (original angle) – cot (final angle)]
Height of the lighthouse = 100 / (cot 30° – cot 45°) = 50 m
Detailed Explanation
Here, we use a specific formula to find the height of the lighthouse. The formula involves the cotangent (cot) of the given angles. The cotangent of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the opposite side. By substituting the angles into the equation and the known distance of the two ships (100 m), we can calculate the height. The resultant calculation shows that the lighthouse height is 50 meters.
Examples & Analogies
Think of using a protractor to measure angles from where you stand on the ground (the boats in our analogy). The mathematical operation of using cotangent effectively helps you calculate how far up you need to go to reach a height that corresponds to that angle, giving you a clear idea of the lighthouse's height above sea level.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Angles of Depression: Important for understanding how to measure heights from a distance
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Cotangent Function: A critical trigonometric function that helps in calculating heights based on angles
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Height Calculation Method: Understanding the formula utilized in determining the heights of objects like lighthouses
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example: A lighthouse's height can be calculated when given the distance between two ships and the angles of depression to each.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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To find a height, angle in sight. Cotangent's your guide, day or night!
📖 Fascinating Stories
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Imagine a lighthouse keeper surveying ships. With angles and distances, he calculates the height with precision—his trusty cotangent is key!
🧠 Other Memory Gems
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Remember HDC: Height = Distance divided by (Cot(Angle1) - Cot(Angle2))!
🎯 Super Acronyms
C.A.D
- Cotangent Angles Distance signifies how we calculate heights!
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Angle of Depression
Definition:
The angle formed between the horizontal line and the line of sight pointing downward to an object.
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Term: Cotangent
Definition:
The ratio of the adjacent side to the opposite side in a right triangle, equivalent to the reciprocal of tangent.
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Term: Height Calculation
Definition:
The process of determining the vertical distance from a point to a designated reference level.