Loading the lesson
Loading the lesson
Search AllRounder.ai
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Learn it your way
A short class with Sarah and her students. The other ways are under More.
Sign up free to listen. Transcripts are free to read.
Today, we are going to explore how to find the height of objects using angles of elevation. Can anyone tell me what an angle of elevation is?
Isn't it the angle between the horizontal line and the line of sight when looking up?
Exactly! When you look up at an object, the angle you create with the horizontal line is the angle of elevation. In our example, the observer looks up at the chimney, measuring at 45 degrees. This helps us calculate its height.
Wait! Why do angles help us measure height?
Great question! We can use trigonometric ratios. In this case, we can set up a right triangle to relate the angles and the distances involved!
So, we can find the height just by knowing the angle and the distance?
Yes! And that’s the power of trigonometry. Let's move on to setting up the right triangle for this scenario.
Now, let's set up our triangle. We have the chimney represented as AB and the observer's height as BE. Who remembers the height of the observer?
The observer is 1.5 meters tall!
That's correct! So, AE will be the height from the observer's line of sight to the top of the chimney, which we need to find. Now, what distance do we have?
The observer is 28.5 meters away from the chimney.
Right! So, we can express AB as AE plus the observer's height. We combine these to find the total height of the chimney. Let’s calculate AE using the tangent function. What is the tangent of 45 degrees?
It’s 1!
Exactly! So we set it up as tan(45°) = AE/DE, which helps us calculate the ratios.
Now we've established that tan(45°) = AE/28.5 m. Since we know that tan(45°) is 1, we can say AE = 28.5 m. Can anyone remind me how to find the total height of the chimney?
We add AE to the height of the observer!
Exactly! So what is AB — the total height of the chimney?
It’s 30 meters because AE is 28.5 meters plus the observer's height of 1.5 meters!
Well done! This demonstrates how we can use trigonometry to calculate heights in real-world situations.
The process of determining the height of an object using the angle of elevation and distance.
Functions like tangent that relate angle measures to side lengths in right triangles.
Understanding the relationship between the sides and angles in right triangles.
An observer standing 1.2 meters tall wants to find the height of a tree 35 meters away with an angle of elevation of 60 degrees. Using the tangent function, we can determine the height.
If a building is observed from a distance of 50 meters with an angle of elevation of 30 degrees, what would be its height using trigonometric ratios?
To find the height of a tower or tree, just look up high and let angles be free.
Once there was a curious observer who hailed from a hill. She measured a chimney tall and got quite a thrill. With angles and distance and her height in the way, she calculated the total height without any delay.
A mnemonic to remember: AE is added to BE for total height, A + B = all right.
Use H = A + B for height calculation: H for height, A for above eyes, B for below.
Term
Angle of Elevation
Click to flip
Term
Tangent Function
Click to flip
The angle between the horizontal and the line of sight when looking up at an object.
A trigonometric ratio used to relate the angle of elevation to the opposite side and adjacent side of a right triangle.
A triangle that has one angle measuring 90 degrees.
A person or entity viewing an object from a specified distance and height.
This section illustrates how to calculate the height of a chimney using trigonometry based on the observer's position and angle of elevation.
In this example, an observer measures the angle of elevation to the top of a chimney from a distance. Using trigonometric functions, specifically the tangent of the angle, we determine the height of the chimney based on the observer's height and distance from the chimney.
In this example, we explore how to determine the height of a chimney based on an observer's distance from it and the angle of elevation to its top. The observer is 1.5 meters tall and positioned 28.5 meters away from the chimney. The angle of elevation, measured from the observer's line of sight, is 45 degrees.
To solve this problem, we recognize that the scenario can be represented as a right triangle where:
Using the tangent ratio, we have the relationship: tan(angle) = opposite/adjacent. Here, tan(45°) = AE / DE. Since tan(45°) = 1, we find that AE = DE = 28.5 m. Combining this with the height of the observer, the total height of the chimney (AB) is calculated as: AB = AE + BE = 28.5 m + 1.5 m = 30 m. Thus, the height of the chimney is 30 meters. This example highlights the practical application of trigonometric ratios in solving real-world measurement problems.