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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 explore how to determine distances using angles of depression. Can anyone remind me what an angle of depression is?
Isn't it the angle below the horizontal line of sight?
Exactly! Now, we have a bridge that's 3 meters high. We'll be looking at two angles of depression from this height: 30° and 45°. How do you think we can find the width of the river using these angles?
We can use right triangles since we know the height and the angles!
Correct! We'll create two right triangles based on these angles. Let's delve into how to calculate the distances.
Using triangle APD where ∠A is 30°, what can we say about the relationship between the sides?
We can use the tangent function for tan(30°) = PD/AD, right?
Exactly! Is anyone able to calculate AD from that formula?
Yes! It's AD = PD * √3. If PD is 3 m, then AD = 3√3 m.
Very well done! Now let’s move to triangle PBD. How can we use the angle of 45°?
Now we have AD calculated. For triangle PBD with an angle of 45°, what can we conclude?
In that case, BD should equal PD because tan(45°) = 1!
Correct again! BD = 3 m. Now, who can tell me the total width of the river AB?
It would be AB = AD + BD which is 3 + 3√3 m!
Exactly! That gives us the final width. Great teamwork, everyone!
Before we finish, let’s recap how we solved the problem. What steps did we take?
We started by identifying angles of depression!
Yes! And then we set up our triangles using tangent ratios based on those angles. But what’s special about the angle of 45°?
It makes the two sides equal in triangle PBD!
Correct! This is an important property of 45° angles. Let's remember these points for future geometry problems.
Consider a building that is 10 meters tall. If the angle of depression to the ground from the top of the building is 60°, what is the distance from the base of the building to the point directly below the top of the building?
From a hill 20 m high, if the angle of depression to a car is 30°, calculate the distance from the base of the hill to the car.
If you see the bank, look down the plank; at thirty degrees you’ll see the trees!
Imagine a bridge where a person looks down, seeing a beautiful river flowing below. The angles guide them to calculate the distance with ease, like a treasure hunt under the trees.
When calculating lengths, remember: TAD (Tangent for Angle and Distance).
Use TAD to remember Tangent, Angle, Distance – the key to solving!
Term
Angle of Depression
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Term
Height of Bridge
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Term
Tangent Function
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The angle formed by a horizontal line and a line of sight from above to a point below the horizontal.
The measure of how tall something is from its base to its top, here referring to the height of the bridge.
A trigonometric function that relates the angle of a right triangle to the ratio of the opposite side to the adjacent side.
This section involves calculating the width of a river using angles of depression from a bridge.
In Example 7, students learn how to determine the width of a river based on angles of depression observed from a bridge. The example illustrates the application of trigonometric ratios in right triangles, specifically using angles of depression and the height of the bridge.
In this example, a person stands on a bridge that is 3 meters high and records the angles of depression to the banks of a river on both sides. The angles of depression are 30° and 45°, respectively. By applying the principles of right triangle trigonometry, we can determine the width of the river by calculating the horizontal distances to each bank from the foot of the bridge.
Definitions and Trigonometric Relationships Used:
In the analysis:
Finally, the total width of the river AB is the sum of AD and BD, yielding the final result in the form of 3(1 + √3)m.