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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 going to learn about angles of depression. Can anyone tell me what an angle of depression is?
Is it the angle formed between a horizontal line and the line of sight looking down?
Exactly, great definition! Remember, as we look down from a height, we create an angle with the horizontal line. We can find some important relationships using these angles.
How do we use those angles to find heights or distances?
We can use trigonometric functions, especially the tangent function which relates angles to opposite and adjacent sides in right-angled triangles. Let's explore how with an example.
In our example, we have two buildings. The first is 8 m tall, and we need to find the height of the multi-storeyed building. What are the angles we are given?
The angles are 30° to the top of the building and 45° to the bottom.
Correct! So, let’s represent these in a diagram to better visualize the problem. Remember, the angle at the top relates to the top of our building and the angle at the bottom relates to the base.
Now, how can we use these angles? Who can set up the equation for the angle of depression of 30° first?
For triangle PBD, we can say tan(30°) = PD/BD, which implies BD = PD√3.
Perfect! Now what about the other triangle, PAC?
Since tan(45°) = 1, PC must equal AC.
Well done! By linking these two triangles, we can solve for both height and distance here.
Now we have two equations. If PD + 8 = BD and BD = PD√3, we can solve for PD. Can anyone help with the calculations?
So, PD + 8 = PD√3, right? Is that rearranging to find PD?
Exactly! What do we get when we isolate PD?
We get PD as a function of 8 and √3!
Right! Don't forget to plug back in to find the height of the multi-storeyed building and the distance. Get ready to wrap this up!
Let's recap. We defined angles of depression, set up our problem, and applied tangent ratios to find our height and distances. What should we remember about using trigonometry in real-world examples?
We can effectively solve for unknown heights and distances given angles!
Exactly! Trigonometry is powerful for solving problems in various fields, from architecture to navigation. Keep practicing!
The angle between the horizontal line and the line of sight down to an object.
A primary trigonometric function used to relate opposite sides and adjacent sides of a triangle.
Utilizing angles and tangent to find unknown measurements in geometric figures.
If you are on top of a 10 m tall building, and you see another building that is 5 m tall, using angles of depression can help calculate their distance apart.
When measuring the height of a tower using the angle of depression from a certain height, you apply tangent ratios to solve for the tower's height.
When looking down with sight so keen, the angle shows the height unseen.
Imagine you are on top of a tall building, peering down at the street below. You measure the angle to the edge of the sidewalk and to the tallest tree nearby. Each angle gives you a clue about how far apart everything is and how high it stands.
Remember 'To Opposite A' which stands for Tangent = Opposite/Adjacent.
HAD = Height and Distance from Angles.
Term
Angle of Depression
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Term
Tangent Function
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Term
Height Calculation
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The angle formed between a horizontal line from an observer's eye and the line of sight to an object below the horizontal level.
In a right-angled triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
A triangle that has one angle equal to 90 degrees.
The measurement of an object from its base to its top.
The space between two points measured along the horizontal plane.
This section explores the angles of depression from the top of a multi-storeyed building to an 8 m tall building, using trigonometry to determine the height of the multi-storeyed building and the distance between the two.
In this example, we analyze angles of depression from the top of a multi-storeyed building to both the bottom and the top of an adjacent 8 m tall building. By applying tangent function properties from right triangles, we derive the heights and distances involved, leading to practical applications of trigonometric concepts.
In Example 6, we are tasked with finding the height of a multi-storeyed building (denoted as PC) and the horizontal distance to an 8 m tall building (denoted as AB). The angles of depression to the top and bottom of building AB from the top of building PC are 30° and 45° respectively. Using properties of alternate angles and tangent ratios for both right triangles (PBD and PAC), we established relationships between the height and distances involved.
From the triangle PBD with angle PBD = 30°, we use the tangent property: tan(30°) = PD/BD, thus establishing that BD = PD√3. For triangle PAC with angle PAC = 45°, we have PC = AC due to the properties of 45° triangles (where the lengths of opposite and adjacent sides are equal).
By combining these relationships, we conclude that the height of the multi-storeyed building is approximately (4√3 + 3) m, and the distance between the two buildings also measures (4√3 + 3) m. This example effectively demonstrates the application of trigonometric functions in solving real-world geometric problems.