9.1.4 - Example 4
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Understanding Angles of Elevation
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Today, we're learning about angles of elevation. When we look at something above us, the angle formed with the horizontal line is called the angle of elevation. Can someone give me an example of where you might see an angle of elevation?
Like looking up at a tall building?
Exactly! Now, if we know the angle and the height of the building, we can find the distance from where we're standing to the building. This uses the tangent function, defined as the opposite over the adjacent side in a right triangle.
So the height of the building is the opposite side, and the distance from the building is the adjacent?
Correct! Let’s remember this with the mnemonic H-O-A (Height-Over-Adjacent) to recall tangent relationships.
Now, can anyone tell me what the tangent of 30° is?
It's 1 over the square root of 3.
That's right! Let's see how we can use this to solve our example.
Finding the Distance PA
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Now that we understand the angle of elevation, let’s calculate PA using the height of the building, which is 10 m. Using the tangent of 30°, which we noted earlier is 1/√3, we set up our equation: tan 30° = AB/PA.
So that means PA equals 10 divided by tan 30°?
Exactly! When you solve that, you will find that PA equals 10√3. What’s the numerical value for √3?
It's approximately 1.732.
Great! So, what is PA then?
That would be around 17.32 m.
Exactly! Fantastic work. This distance helps us understand how far away the point P is from the building.
Calculating Flagstaff Height
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Next, let’s find the height of the flagstaff. Who remembers what we need to do next?
We need to form a new triangle using the flag and the angle from P.
Exactly! We know the angle of elevation to the top of the flagstaff is 45°. This means our triangle includes AD, which is AB plus the height of the flag (DB).
So, if AD = 10 + DB and we use tan 45°!
Correct! And since tan 45° equals 1, we can set up our next equation. Can anyone show how to simplify?
So it's 10 + x = PA. Remember, PA is around 17.32 m.
Thus, what do we find x to be?
The height of the flagstaff x is 7.32 m.
Awesome! We worked it out together. You've all done great in understanding this example.
Introduction & Overview
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Quick Overview
Standard
In this example, we use angles of elevation to find the height of a flagstaff atop a 10 m building and the distance from a point on the ground to the building. Utilizing right triangles and the tangent function, we derive both measurements.
Detailed
In this section, Example 4 illustrates a practical application of trigonometry using angles of elevation to find unknown distances and heights. Starting with a point P on the ground, where the angle of elevation to the top of a 10 m tall building (AB) is measured at 30°, the distance from point P to the building's base (PA) is calculated using the tangent function. Subsequently, the height of a flagstaff positioned atop the building (BD) is determined with the angle of elevation to the flag's top (AD) measured at 45°. By creating two right triangles, the solution employs the relationships of tangent and height to arrive at the required distances. The example emphasizes the practical use of trigonometry in real-life scenarios.
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Understanding the Scenario
Chapter 1 of 3
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Chapter Content
From a point P on the ground the angle of elevation of the top of a 10 m tall building is 30°. A flag is hoisted at the top of the building and the angle of elevation of the top of the flagstaff from P is 45°.
Detailed Explanation
In this scenario, we are given a situation where we have a point P on the ground and two points of interest: the top of a building and the top of a flagstaff that is mounted on the building. The height of the building is known to be 10 meters, and the angles of elevation from point P to these points are also provided. The angle of elevation to the top of the building is 30 degrees, and to the top of the flagstaff, it is 45 degrees. This scenario uses trigonometric concepts to help us find two unknowns: the length of the flagstaff and the distance from the point P to the foundation of the building.
Examples & Analogies
Imagine you are standing a certain distance away from a tall tree (the building) and you're looking up at the top of it. The angle you look up at to see the top of the tree is like the 30 degrees mentioned. Now, if there’s a flag on this tree that’s taller than the tree itself, the angle you look up to see the flag is sharper, like 45 degrees. This helps us visualize how trigonometry is used to calculate heights and distances in real life.
Calculating Distance to the Building
Chapter 2 of 3
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Chapter Content
We have tan 30° = AB / AP, i.e., 1/√3 = 10 / AP. Therefore, AP = 10√3, i.e., the distance of the building from P is 10√3 m = 17.32 m.
Detailed Explanation
To find the distance from point P to the base of the building (denoted as AP), we apply the tangent function, which relates the angle of elevation to the opposite side (height of the building, AB) and the adjacent side (distance from the point to the base of the building). For an angle of 30 degrees, the tangent value is 1/√3. We set up the equation as 1/√3 = 10/AP. By cross-multiplying and solving for AP, we derive AP = 10√3 meters. When calculated, this gives us approximately 17.32 meters as the distance from point P to the building.
Examples & Analogies
Think of it like using a ladder to reach different heights. For a tree that’s 10 meters tall, if you stand a little more than 17 meters away (measured on the ground), the angle you look up at the top of the tree would be 30 degrees. This helps us visualize the relationship between height and distance when measuring from the ground.
Determining the Length of the Flagstaff
Chapter 3 of 3
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Chapter Content
Now, let us suppose DB = x m. Then AD = (10 + x) m. In right △PAD, tan 45° = AD / AP, therefore, 1 = 10 / (10√3 + x). Thus, x = 10(√3 - 1) = 7.32 m.
Detailed Explanation
To find the length of the flagstaff (DB), we designate it as 'x'. The total height to the top of the flagstaff (AD) can be expressed in terms of the height of the building plus the length of the flagstaff, resulting in (10 + x) meters. In triangle PAD, we again utilize the tangent function, where the angle of elevation is 45 degrees. For this angle, the tangent value is 1, meaning the opposite side (AD) equals the adjacent side (AP). We set up the equation 1 = 10/(10√3 + x), leading us to find x. Simplifying, we find the length of the flagstaff as approximately 7.32 meters.
Examples & Analogies
Imagine the flagstaff is like an extra pole you add to the top of the tree. The relationship between the height of the pole and the distance from which you stand mirrors how we calculate the extra height if you know how high the tree trunk is. So, if the pole is around 7.32 meters, you can visualize how much taller the entire structure is when standing back at the calculated distance.
Key Concepts
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Angle of Elevation: The angle between the horizontal line and the line of sight to an object above.
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Tangent Function: A crucial concept in trigonometry representing the ratio of the opposite side over the adjacent side in a right triangle.
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Opposite and Adjacent Sides: Terms that describe the relationship between the sides of a right triangle in relation to a specific angle.
Examples & Applications
Using a 10 m tall building, with an angle of elevation of 30°, we find the distance from point P to the building.
From the same point, with a 45° angle of elevation to the flagstaff atop the building, we calculate the flagstaff's height.
Memory Aids
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Rhymes
Up to the sky, the angle is nigh, check the height from the ground, let it be found!
Stories
Once, a student looked up at a tall building, curious about its height. With a straight line of sight, they measured the angle, finding the distant ground, creating triangles in the air.
Memory Tools
H-O-A: Height over Adjacent for Tangent’s game!
Acronyms
A for Angle, E for Elevation, T for Tangent — remember AET to keep it straight!
Flash Cards
Glossary
The angle formed between the horizontal line from the observer and the line of sight to a point above.
A trigonometric function defined as the ratio of the length of the opposite side to the length of the adjacent side in a right triangle.
In a right triangle, the side opposite the angle of interest.
In a right triangle, the side adjacent to the angle of interest.
A triangle where one of the angles is a right angle (90 degrees).
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