Enrol to start learning
Reading is open to everyone. Enrolling is free, and it is what unlocks the audio lessons, practice tests and progress tracking.
5.4. Applications of Trigonometry
Interactive Audio Lesson
Unlock the classroom podcast
The transcript is above and free to read. A free account plays the conversation back.
Create a free account
Today, we'll learn about angles of elevation. This is the angle formed from the horizontal line when you look up towards an object.
Why do we need to use trigonometry for angles of elevation?
Great question! We use trigonometry to calculate distances and heights that are hard to measure directly. For example, by knowing the height of a tower and measuring the angle of elevation from the ground, we can find how far away we are from the tower.
Can you give an example of that?
Sure! If a tower is 50 m tall and the angle of elevation is 30 degrees, we can use the tangent function to find the distance from the tower.
I remember that tan(θ) = opposite/adjacent!
Exactly! And in our example, the 'opposite' is the height of the tower. So, we set it up as tan(30°) = 50/distance.
And then we can solve for distance, right?
That's right! Let’s work on solving that together.
Unlock the classroom podcast
The transcript is above and free to read. A free account plays the conversation back.
Create a free account
Continuing from our last discussion, if we apply the tangent equation, what do we find for distance?
We have tan(30°) = 50/distance, so distance = 50/tan(30°).
Correct! And what is tan(30°)? Do you remember?
Yes, it’s 1/sqrt(3) or about 0.577!
Excellent! Now plug that value into the distance equation.
So distance = 50/(1/sqrt(3)), which multiplies to 50*sqrt(3).
Which is approximately 86.6 m!
Fantastic! So this process can be used in various applications to determine distances.
Unlock the classroom podcast
The transcript is above and free to read. A free account plays the conversation back.
Create a free accountNow, let's explore angles of depression. This is the angle formed when looking down from a certain height.
So, if we're at the top of a building looking down, that’s an angle of depression?
Correct! Just like angles of elevation, we can also use them in calculations.
Can we use the same trigonometric functions?
Yes, exactly! You’ll often find that you can still apply sin, cos, or tan based on the context.
Can you show an example of this?
Certainly! Let’s say you are on a cliff looking down at a boat with an angle of depression of 45 degrees. You can use similar tangent relationships to calculate the distance from the cliff you are at!
So it works the same as before?
Exactly! It’s just the idea of looking upward versus downward.
Overview
Short Summary
This section explores the practical applications of trigonometry in measuring heights and distances.
Medium Summary
In this section, we delve into how trigonometry is applied to determine heights and distances that are challenging to measure directly, as well as angles of elevation and depression. Practical examples illustrate these concepts in action.
Detailed Summary
Detailed Summary
In this section, we discuss the real-world applications of trigonometry, focusing on its utility in calculating heights and distances that are difficult to measure directly. Trigonometry helps us find angles of elevation and depression that are essential in various fields such as architecture, navigation, and surveying. An example illustrates the process of determining the distance to a tower by measuring the angle of elevation from a certain point on the ground and using the height of the tower. This section serves as a bridge between theoretical concepts and their practical implementations, demonstrating the importance of trigonometry in everyday life.
Reference YouTube Videos
Audio Book
Unlock the audio lesson
The script is above and free to read. A free account plays it back, in the voice you pick.
Create a free accountTrigonometry is used to find: ● Heights and distances that are difficult to measure directly. ● Angles of elevation and depression.
Detailed Explanation
In this chunk, we learn about the practical uses of trigonometry. One of the main applications is in measuring heights and distances that we cannot easily access. For example, if you want to know the height of a tree but cannot climb it, trigonometry provides a method to calculate that height using angles. Additionally, trigonometry helps in understanding angles of elevation (looking up) and depression (looking down).
Examples & Analogies
Imagine you are standing some distance away from a tall building. You want to know how tall the building is, but it’s too unsafe to go near it or climb to the top. By measuring the angle at which you are looking up to the top of the building (the angle of elevation), you can use trigonometry to calculate the building’s height with just this angle and your distance from the base.
Unlock the audio lesson
The script is above and free to read. A free account plays it back, in the voice you pick.
Create a free account✦ Example: From a point on the ground, the angle of elevation to the top of a tower is 30°. If the tower is 50 m high, find the distance of the point from the base of the tower. Solution: Let distance = d m. tan 30° = Height / Distance = 50 / d tan 30° = 1 / √3 ⇒ 1 / √3 = 50 / d d = 50√3 ≈ 86.6 m.
Detailed Explanation
In this example, we learn how to apply the concept of an angle of elevation to find a distance. The angle of elevation to the top of a tower is given as 30°, while the height of the tower is 50 meters. We denote the horizontal distance from the tower as 'd'. Using the trigonometric function tangent (tan), which relates the angle with the opposite side (height of the tower) and adjacent side (distance from the tower), we set up the equation. The tangent of 30° equals the height (50 m) divided by this unknown distance (d). Solving the equation gives us the distance d as approximately 86.6 meters.
Examples & Analogies
Think about standing on a flat ground looking up at a tall structure, such as a cell tower. If you hold a protractor to measure the angle you need to look up to see the top of the tower and you know how high the tower is, you can use trigonometry to find out how far away from the tower you are standing. This is similar to determining how far back you need to stand to get a clear view of a mountain while ensuring you don’t get too close!
--
Key Concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
Trigonometry: The branch of mathematics that deals with the relationships between the sides and angles of right-angled triangles.
Angle of Elevation: The angle formed when looking up towards an object.
Angle of Depression: The angle formed when looking down towards an object.
Applications: Trigonometry is useful in various fields such as surveying, architecture, and engineering.
Examples
Step-by-step examples to apply the section's ideas and test your understanding.
To find the distance from a tower given its height and the angle of elevation, use the formula distance = height/tan(angle). For instance, if a tower is 50m tall and the angle of elevation is 30°, the distance would be approximately 86.6 m.
During construction, architects may need to know the angle of elevation from a ground point to a structure to ensure proper height and alignment.
Memory Aids
Interactive tools to help you remember key concepts
Stories
Memory Tools
Flash Cards
Glossary
Angle of Elevation
The angle formed from the horizontal line of sight when looking upward towards an object.
Angle of Depression
The angle formed from the horizontal line of sight when looking downward towards an object.
Tangent (tan)
A trigonometric function that relates the angle of a triangle to the ratio of the length of the opposite side to the length of the adjacent side.