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A short class with Sarah and her students. The other ways are under More.
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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.
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.
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.
The angle between the horizontal line and the line of sight to an object above.
A crucial concept in trigonometry representing the ratio of the opposite side over the adjacent side in a right triangle.
Terms that describe the relationship between the sides of a right triangle in relation to a specific angle.
Up to the sky, the angle is nigh, check the height from the ground, let it be found!
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.
H-O-A: Height over Adjacent for Tangent’s game!
A for Angle, E for Elevation, T for Tangent — remember AET to keep it straight!
Term
Angle of Elevation
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Term
Tangent Function
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Term
Height of Building
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Term
Distance PA
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Term
Height of Flagstaff
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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).
This section involves calculating the height of a flagstaff and the distance from a point to a building using trigonometric relationships.
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.
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.