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Interactive Audio Lesson
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Understanding Angles of Elevation
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Today we're learning about angles of elevation. Can anyone explain what an angle of elevation is?
Isn't it the angle formed from the horizontal line up to an object?
Exactly! When you look up at something, like the top of a tower, the angle you measure is the angle of elevation. Can someone give me an example where we might use this in real life?
Like when measuring how tall a building is from the ground?
Great example! That's just what we're going to do today. We'll find the height of a tower using the angle of elevation and some basic trigonometry.
Introduction to Trigonometric Ratios
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Now, let's talk about the trigonometric ratio we will use. Who can tell me what tan(ΞΈ) relates to in our triangle?
It's the opposite side over the adjacent side!
Correct! So in our problem with the tower, 'AB' is the height, which is opposite, and 'BC' is the distance from the tower on the ground, which is adjacent. If I say tan(60Β°), what does that equal using our sides?
tan(60Β°) = AB / 15?
Exactly! So we can set up the equation tan(60Β°) = AB/15.
Calculating the Height of the Tower
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Now we know tan(60Β°) is equal to β3. Letβs substitute that into our equation. Who can help me out?
So, tan(60Β°) = β3 = AB / 15!
Perfect! Now letβs solve for AB.
We multiply both sides by 15, so AB = 15β3!
Exactly! And that means the height of the tower is 15β3 meters. Can anyone tell me what this implies?
It means towers of different heights can be calculated from a distance, depending on the angle of elevation!
Right! Understanding angles of elevation and using trig ratios helps us in many fields, including architecture and engineering.
Review and Application
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Now that we've calculated the height of the tower, letβs summarize our steps. What did we determine first?
We identified the triangle and labeled the sides based on the angle of elevation!
Great! And then how did we apply the tangent function?
We used tan(60Β°) = AB / 15 to find the height.
Exactly! Always remember, identifying the components of a triangle is crucial. Any questions before we finish today?
I have a question about what would happen if the distance was different!
We'll explore that scenario in our next lesson. Excellent work today, everyone!
Introduction & Overview
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Quick Overview
Standard
In this section, students learn how to determine the height of a tower based on the distance from the tower and the angle of elevation using basic trigonometric ratios, specifically the tangent function. The problem is solved step-by-step, showcasing the application of tan(60Β°).
Detailed
In Example 1, we are tasked with finding the height of a vertical tower from a given point on the ground, situated 15 meters away from the base. An angle of elevation of 60Β° is observed towards the top of the tower, prompting us to apply trigonometric ratios. The key concept here is the tangent function in a right-angled triangle, which helps relate the angle of elevation to the opposite side (height of the tower, denoted as AB) and the adjacent side (distance from the point to the tower, denoted as BC). Utilizing the ratio tan(60Β°) = AB / BC, we substitute the value of BC (15m) and perform the necessary calculations to derive the height of the tower (15β3 m), reinforcing the significance of understanding angles and right triangles in practical applications.
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Audio Book
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Problem Introduction
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A tower stands vertically on the ground. From a point on the ground, which is 15 m away from the foot of the tower, the angle of elevation of the top of the tower is found to be 60Β°. Find the height of the tower.
Detailed Explanation
In this problem, we have a vertical tower and a point on the ground 15 meters away from it. The angle of elevation means how high from the horizontal line the top of the tower appears to the observer at that point. Here, the angle of elevation to the top of the tower is 60 degrees, and we need to calculate the height of the tower, which we will refer to as 'AB'.
Examples & Analogies
Think of standing at a distance from a tall building. When you look up at the top of the building, you can imagine drawing an angle from your eyes to the top of the building; this is similar to the angle of elevation in the problem.
Diagram Representation
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First let us draw a simple diagram to represent the problem (see Fig. 9.4). Here AB represents the tower, CB is the distance of the point from the tower and angle ACB is the angle of elevation.
Detailed Explanation
To visualize the problem, we can draw a right triangle. In this triangle, one side (AB) represents the height of the tower, while the other side (BC) represents the horizontal distance from the observer to the tower (15 m). The angle ACB is the angle of elevation (60Β°). The triangle formed helps us to apply trigonometry to find the height of the tower.
Examples & Analogies
Imagine a ladder leaning against a wall. The height the ladder reaches on the wall is like the height of the tower, and the distance the base of the ladder is from the wall is like the 15 m in our problem.
Triangle Identification
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Also, ACB is a triangle, right-angled at B. To solve the problem, we choose the trigonometric ratio tan 60Β° (or cot 60Β°), as the ratio involves AB and BC.
Detailed Explanation
The triangle ACB is identified as a right triangle because one angle measures 90 degrees at point B. Here, we will use the tangent function, which relates the opposite side (height of the tower, AB) to the adjacent side (distance to the tower, BC). The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.
Examples & Analogies
If youβve ever used a ladder, think about how when you measure the height you reach on a wall (the height of the ladder against the wall, here represented by AB), you're finding a relationship between how high you can reach and how far the base of the ladder is from the wall (like BC).
Trigonometric Calculation
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Now, tan 60Β° = AB / BC i.e., 3 = AB / 15 i.e., AB = 15 * 3.
Detailed Explanation
Using the tangent function for 60 degrees, we know tan 60Β° = β3 (or approximately 1.732). In this case, we set up the equation as tan 60Β° = AB / 15. By multiplying both sides by 15, we find that the height of the tower AB equals 15 times tan 60Β°, which simplifies to AB = 15β3.
Examples & Analogies
This calculation is similar to determining how much higher you would reach when standing on a hill. The distance to the top of the tower corresponds to the height of the hill. The steeper the hill (or the greater the angle), the higher the height you could potentially reach.
Conclusion
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Hence, the height of the tower is 15β3 m.
Detailed Explanation
After performing the calculations, we conclude that the height of the tower (AB) is 15β3 meters. This means that the actual height of the tower is approximately 25.98 meters when calculated numerically because the value of β3 is about 1.732.
Examples & Analogies
This conclusion can be visualized as accurately measuring how tall a building is compared to how far away you are. It's important because it shows how mathematics can be applied to real-world situations like measuring objects at a distance.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Angle of Elevation: The angle from the observer's horizontal line of sight up to an object.
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Right Triangle: A triangle with one angle equal to 90Β°.
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Tangent: A trigonometric function relating opposite and adjacent sides in a right triangle.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example of measuring the height of a building or tree using its angle of elevation from a certain distance.
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Finding the height of a ladder leaned against a wall by forming a right triangle.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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If you want to find the height, with a glance up to the light, use tan to find the ratio right!
π Fascinating Stories
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Imagine standing far from a tall tower, looking up as its peak begins to flower, using angles to measure height like a shower.
π§ Other Memory Gems
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To remember tan, think 'Tall And Now' for the opposite over adjacent!
π― Super Acronyms
T.O.A.
- Tangent = Opposite over Adjacent.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Angle of Elevation
Definition:
The angle formed by a horizontal line to an observer's line of sight upward to an object.
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Term: Trigonometric Ratio
Definition:
A ratio of two sides of a right triangle, commonly used in trigonometry, such as sine, cosine, or tangent.
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Term: Tangent Function (tan)
Definition:
A function that relates the angle ΞΈ in a right triangle to the ratio of the length of the opposite side to the length of the adjacent side.