Angles of Elevation and Depression
Subject : Mathematics
Class : JSS 2
Term : Second Term
Week : Week 8
Topic : Angles of Elevation and Depression
Previous Lesson :
Trigonometry and Euclidean Geometry ( Pythagorean Theorem)
Objectives:
- Define angles of elevation and depression
- Understand how to measure angles of elevation and depression
- Solve problems involving angles of elevation and depression
- Apply angles of elevation and depression to real-life situations
Materials:
- Whiteboard and markers
- Protractor and clinometer
- Practice problems handout
Content :
Angles of Elevation and Depression
Angles of elevation and depression are important concepts in mathematics and they are used to describe the angles formed between a horizontal line and a line of sight to an object.
To help you understand this concept better, let’s take a look at some examples:
- Angles of Elevation:
An angle of elevation is the angle formed when an observer looks up at an object from a horizontal line. For example, if you are standing on the ground and looking up at a bird flying in the sky, the angle between the horizontal line and your line of sight to the bird is the angle of elevation.
Let’s say that you are standing on the ground and you look up at the top of a tree. The angle of elevation is the angle between the horizontal line and your line of sight to the top of the tree.
- Angles of Depression:
An angle of depression is the angle formed when an observer looks down at an object from a horizontal line. For example, if you are standing on a bridge and looking down at a boat in the water, the angle between the horizontal line and your line of sight to the boat is the angle of depression.
Let’s say that you are standing on the top of a building and looking down at the street below. The angle of depression is the angle between the horizontal line and your line of sight to the street below.
So, in summary, angles of elevation are angles formed when an observer looks up at an object from a horizontal line, while angles of depression are angles formed when an observer looks down at an object from a horizontal line.
What is an angle
In mathematics, an angle is the figure formed by two rays, or straight lines, that share a common endpoint called the vertex. The rays are called the sides or legs of the angle. Angles are measured in degrees and are used to describe the amount of rotation between two lines or objects. The degree of an angle is determined by the amount of rotation needed to bring one of its sides to coincide with the other side. Angles can be acute (less than 90 degrees), right (exactly 90 degrees), obtuse (greater than 90 degrees but less than 180 degrees), straight (exactly 180 degrees), reflex (greater than 180 degrees), or full (exactly 360 degrees). Angles are an important concept in mathematics and are used in many applications, including geometry, trigonometry, physics, and engineering
How angles are measured
When we talk about measuring angles, we use a unit of measurement called degrees. Degrees are used to describe the size of an angle.
To measure an angle in degrees, we start by drawing two lines that meet at a point to form a corner. This point is called the vertex of the angle. Then, we place a protractor on one of the lines, making sure that the base of the protractor lines up with the vertex of the angle.
The protractor has a half-circle shape with numbers marked along the curved edge from 0 to 180 degrees. To measure the angle, we look at the other line and find where it intersects with the curved edge of the protractor. The number that this line intersects with is the measure of the angle in degrees.
For example, if the other line intersects with the curved edge of the protractor at the number 50, then the measure of the angle is 50 degrees.
It’s important to note that angles can also be larger than 180 degrees. When this happens, we use a full circle, which is 360 degrees, to measure the angle. For example, a full turn around a circle is 360 degrees
Angles of elevation and depression are terms used in mathematics to describe the angles formed between a horizontal line and a line of sight to an object.
- An angle of elevation is the angle formed when an observer looks up at an object from a horizontal line. It is the angle between the horizontal line and the line of sight to the object above the horizontal line.
- An angle of depression, on the other hand, is the angle formed when an observer looks down at an object from a horizontal line. It is the angle between the horizontal line and the line of sight to the object below the horizontal line.
These angles are often used in real-life situations, such as in navigation, engineering, and surveying, to help determine the height or distance of an object from a given point. By measuring the angles of elevation or depression, we can use trigonometry to calculate the height or distance of an object
Angles of elevation and depression are two important types of angles used in mathematics to describe the angles formed between a horizontal line and a line of sight to an object.
An angle of elevation is the angle formed when an observer looks up at an object from a horizontal line. For example, if you are standing on the ground and looking up at the top of a tree or a tall building, the angle between the horizontal line and your line of sight to the top of the tree or building is the angle of elevation.
An angle of depression, on the other hand, is the angle formed when an observer looks down at an object from a horizontal line. For example, if you are standing on the top of a building and looking down at the street below, the angle between the horizontal line and your line of sight to the street below is the angle of depression.
These angles are important in various fields such as architecture, engineering, and surveying. By measuring the angles of elevation or depression, we can use trigonometry to calculate the height, distance or depth of an object. For example, if we know the angle of elevation to the top of a building and the distance from the building to our position, we can use trigonometry to calculate the height of the building. Similarly, if we know the angle of depression to the bottom of a well and the distance from the well to our position, we can use trigonometry to calculate the depth of the well
Real life problems on Angles of Elevation and Depression
- Navigation: When sailing or flying, pilots and navigators use angles of elevation to help determine their position and the height of objects around them. For example, if a pilot sees a landmark in the distance and measures the angle of elevation to the landmark, they can use trigonometry to calculate the distance to the landmark.
- Architecture: Architects use angles of elevation to design buildings and structures that fit into the surrounding environment. For example, if an architect is designing a building on a hill, they need to measure the angles of elevation to the hill from different points to ensure that the building fits harmoniously with the slope of the hill.
- Surveying: Surveyors use angles of elevation and depression to measure the height and distance of objects such as buildings, mountains, and trees. For example, if a surveyor needs to measure the height of a tree, they can use a clinometer to measure the angle of elevation to the top of the tree and then use trigonometry to calculate the height of the tree.
- Sports: Athletes use angles of elevation and depression in sports such as archery, golf, and javelin throwing to calculate the trajectory of their projectiles. For example, a golfer needs to calculate the angle of elevation and distance to the hole to make a successful shot.
- Photography: Photographers use angles of elevation and depression to frame their shots and create interesting compositions. For example, a photographer might use a high angle of elevation to take a photo of a cityscape from a skyscraper or a low angle of depression to take a photo of a flower from ground level
Evaluation
- What is an angle of elevation? A. The angle formed when an observer looks down at an object from a horizontal line B. The angle formed when an observer looks up at an object from a horizontal line C. The angle formed between two intersecting lines D. The angle formed between a straight line and a curve
- What is an angle of depression? A. The angle formed when an observer looks down at an object from a horizontal line B. The angle formed when an observer looks up at an object from a horizontal line C. The angle formed between two intersecting lines D. The angle formed between a straight line and a curve
- How is an angle of elevation measured? A. With a ruler B. With a compass C. With a protractor D. With a calculator
- How is an angle of depression measured? A. With a ruler B. With a compass C. With a protractor D. With a calculator
- In which field is the concept of angles of elevation and depression used? A. Mathematics B. Biology C. Geography D. History
- What is a clinometer used for? A. To measure angles of elevation and depression B. To measure the length of a curve C. To measure the area of a triangle D. To measure the volume of a solid
- Which type of angle is larger than 180 degrees? A. Acute angle B. Right angle C. Obtuse angle D. Reflex angle
- Which type of angle is exactly 90 degrees? A. Acute angle B. Right angle C. Obtuse angle D. Reflex angle
- Which of the following is an example of a real-life situation where angles of elevation and depression are used? A. Playing video games B. Watching TV C. Sailing D. Cooking
- What is the degree of a full turn around a circle? A. 90 degrees B. 180 degrees C. 270 degrees D. 360 degrees
Lesson Presentation
Revision
Introduction (10 minutes):
- Introduce the topic of angles of elevation and depression
- Define what angles are in mathematics
- Explain the importance of angles of elevation and depression in real-life situations
Instruction (30 minutes):
- Define angles of elevation and depression
- Demonstrate how to measure angles of elevation and depression using a protractor and clinometer
- Provide examples of problems involving angles of elevation and depression and demonstrate how to solve them
- Discuss real-life situations where angles of elevation and depression are used
Practice (20 minutes):
- Distribute practice problems handout to students
- Have students work in pairs to solve the problems
- Monitor and assist students as needed
Closure (10 minutes):
- Review key concepts learned during the lesson
- Discuss how angles of elevation and depression can be applied to real-life situations
- Encourage students to practice and apply what they have learned in their own lives
Assessment:
- Evaluate students’ understanding of angles of elevation and depression based on their participation in class discussions and their ability to solve practice problems.
Extension:
- Provide additional practice problems for students who have mastered the concepts
- Challenge students to find examples of angles of elevation and depression in their own lives and present them to the class
Weekly Assessment /Test
- What is an angle of elevation? A. The angle formed when an observer looks down at an object from a horizontal line B. The angle formed when an observer looks up at an object from a horizontal line C. The angle formed between two intersecting lines D. The angle formed between a straight line and a curve
- What is an angle of depression? A. The angle formed when an observer looks down at an object from a horizontal line B. The angle formed when an observer looks up at an object from a horizontal line C. The angle formed between two intersecting lines D. The angle formed between a straight line and a curve
- How is an angle of elevation measured? A. With a ruler B. With a compass C. With a protractor D. With a calculator
- How is an angle of depression measured? A. With a ruler B. With a compass C. With a protractor D. With a calculator
- What is the degree of a right angle? A. 45 degrees B. 90 degrees C. 135 degrees D. 180 degrees
- Which of the following is an example of a real-life situation where angles of elevation and depression are used? A. Playing video games B. Watching TV C. Sailing D. Cooking
- What is a clinometer used for? A. To measure angles of elevation and depression B. To measure the length of a curve C. To measure the area of a triangle D. To measure the volume of a solid
- Which type of angle is exactly 180 degrees? A. Acute angle B. Right angle C. Obtuse angle D. Straight angle
- Which type of angle is greater than 180 degrees? A. Acute angle B. Right angle C. Obtuse angle D. Reflex angle
- If the angle of elevation to the top of a building is 45 degrees and the distance from the building to the observer is 50 meters, what is the height of the building? A. 50 meters B. 71 meters C. 100 meters D. 141 meters