Lesson Note on Right-Angled Triangle: Angles of Elevation and Depression for SSS 2

Class: Senior Secondary School 2 (SS2 / SSS 2)
Term: 3rd Term
Week: 9 – 10
Age: 16 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Geometry
Previous Lesson: Cumulative Frequency: Formula Method for Median and Percentiles.
Topic: THE RIGHT-ANGLE TRIANGLE
Subject Matter: Angles of elevation and depression

Specific Objectives

By the end of the lesson, pupils should be able to:

Cognitive Domain:

• Define angle of elevation.
• Define angle of depression.
• Identify angles of elevation and depression in given diagrams.
• Apply trigonometric ratios (sine, cosine, tangent) to solve problems involving angles of elevation and depression.

Affective Domain:

• Appreciate the real-world applications of angles of elevation and depression.
• Show interest in solving problems related to heights and distances.

Psychomotor Domain:

• Draw accurate diagrams representing angles of elevation and depression from word problems.
• Solve practical problems involving angles of elevation and depression using appropriate trigonometric ratios.

Social Domain:

• Collaborate with peers to discuss and solve problems involving angles of elevation and depression.
• Communicate their problem-solving steps clearly to others.

Reference Materials

The following resources were used in planning this lesson:

• Senior Secondary Schools Education Curriculum
• State Unified Scheme of Work
• New General Mathematics for Senior Secondary Schools 2
• https://www.mathsisfun.com/geometry/trigonometry.html
• https://www.dummies.com/education/math/geometry/angles-of-elevation-and-depression/

Instructional Materials

The teacher will teach this lesson with the aid of:

• A tree or a tall object in the school compound
• A desk to represent varying heights
• Measuring tape
• Diagrams illustrating angles of elevation and depression
• Protractor and ruler
• Chalkboard or whiteboard

Rationale for the Lesson

This lesson helps pupils understand how to measure heights and distances indirectly using angles. It connects mathematical concepts to practical situations encountered in everyday life, such as surveying and navigation.

Prerequisite/Previous Knowledge

Pupils are expected to have prior knowledge of right-angled triangles, Pythagoras’ theorem, and basic trigonometric ratios (SOH CAH TOA).

Lesson Content/Board Summary

THE RIGHT-ANGLE TRIANGLE

Angles of Elevation and Depression

In trigonometry, angles of elevation and depression are used to describe the angles between a horizontal line and the line of sight to an object.

The following are key definitions:

• Angle of Elevation: This is the angle formed between the horizontal line and the line of sight when an observer looks upwards at an object. The object is above the horizontal level of the observer.
• Angle of Depression: This is the angle formed between the horizontal line and the line of sight when an observer looks downwards at an object. The object is below the horizontal level of the observer.

Relationship: The angle of elevation from point A to point B is equal to the angle of depression from point B to point A, assuming the same horizontal line. This is because they are alternate interior angles when the horizontal lines are parallel.

Solving Problems Using Trigonometric Ratios

Problems involving angles of elevation and depression often require the use of trigonometric ratios (sine, cosine, and tangent) in right-angled triangles.

• SOH: Sine (Opposite / Hypotenuse)
• CAH: Cosine (Adjacent / Hypotenuse)
• TOA: Tangent (Opposite / Adjacent)

Steps for Solving Problems:

1. Draw a clear diagram representing the problem. Label all knowns and unknowns.
2. Identify the right-angled triangle(s) in the diagram.
3. Determine whether the angle is an angle of elevation or depression.
4. Choose the appropriate trigonometric ratio (sine, cosine, or tangent) based on the known and unknown sides relative to the given angle.
5. Set up the equation and solve for the unknown quantity.
6. State the answer with appropriate units.

Worked Example 1 (Angle of Elevation):

A ladder is leaning against a vertical wall. The base of the ladder is 5 meters from the wall. If the ladder reaches a height of 12 meters on the wall, calculate the angle of elevation of the top of the ladder from the ground.

Solution:

Step 1: Draw a diagram. (A right-angled triangle is formed by the wall, the ground, and the ladder).

Step 2: Identify knowns and unknowns.

• Opposite side (height on wall) = 12 m
• Adjacent side (distance from wall) = 5 m
• Angle of elevation = θ (unknown)

Step 3: Choose the appropriate ratio. Since Opposite and Adjacent are known, use Tangent.

Step 4: Set up the equation and solve.

tan θ = Opposite / Adjacent

tan θ = 12 / 5

tan θ = 2.4

θ = tan⁻¹(2.4)

θ ≈ 67.38°

Therefore, the angle of elevation is approximately 67.4° (to 1 decimal place).

Worked Example 2 (Angle of Depression):

A man standing on top of a 50-meter-tall building observes a car on the ground at an angle of depression of 30°. How far is the car from the base of the building?

Solution:

Step 1: Draw a diagram. (A right-angled triangle is formed by the building, the ground, and the line of sight to the car. The angle of depression is outside this triangle, but its alternate interior angle is the angle of elevation from the car to the man).

Step 2: Identify knowns and unknowns.

• Height of building (Opposite side to the angle of elevation) = 50 m
• Angle of depression = 30°
• Angle of elevation from car to man = 30° (alternate interior angle)
• Distance of car from building (Adjacent side) = x (unknown)

Step 3: Choose the appropriate ratio. Since Opposite and Adjacent are involved, use Tangent.

Step 4: Set up the equation and solve.

tan (angle of elevation) = Opposite / Adjacent

tan 30° = 50 / x

x = 50 / tan 30°

x = 50 / 0.5774

x ≈ 86.60 m

Therefore, the car is approximately 86.6 meters from the base of the building.

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher asks pupils if they have ever looked up at a tall building or down from a high place. The teacher then relates these actions to the concept of angles formed by looking up or down.
Pupils’ Activity: Pupils share their experiences of looking up or down and listen attentively.
Learning Point: Pupils are introduced to the idea of angles related to observation points.

Step 2: Explanation of Angle of Elevation

Time: 7 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher defines the angle of elevation and uses an instructional material (e.g., pointing to the top of a tree or a high point in the classroom) to demonstrate it. The teacher draws a simple diagram on the board to illustrate.
Pupils’ Activity: Pupils listen to the definition, observe the demonstration, and copy the definition and diagram into their notebooks.
Learning Point: Pupils understand and can identify the angle of elevation.

Step 3: Explanation of Angle of Depression

Time: 7 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher defines the angle of depression and uses an instructional material (e.g., looking down from a desk or a window) to demonstrate it. The teacher draws a simple diagram on the board, showing the horizontal line and the line of sight.
Pupils’ Activity: Pupils listen to the definition, observe the demonstration, and copy the definition and diagram into their notebooks.
Learning Point: Pupils understand and can identify the angle of depression.

Step 4: Relationship between Angles of Elevation and Depression

Time: 5 minutes
Teaching Skill: Explanation/Questioning
Teacher’s Activity: The teacher draws a combined diagram showing both an angle of elevation and an angle of depression between two points. The teacher explains that these angles are equal due to being alternate interior angles formed by parallel horizontal lines.
Pupils’ Activity: Pupils observe the diagram, ask questions for clarity, and note the relationship.
Learning Point: Pupils understand that the angle of elevation from one point to another is equal to the angle of depression from the second point to the first.

Step 5: Application of Trigonometric Ratios

Time: 8 minutes
Teaching Skill: Review/Guidance
Teacher’s Activity: The teacher briefly reviews the three basic trigonometric ratios (SOH CAH TOA) and guides pupils on how to choose the correct ratio when solving problems involving angles of elevation and depression. The teacher explains the steps for problem-solving.
Pupils’ Activity: Pupils recall the trigonometric ratios and pay attention to the problem-solving steps.
Learning Point: Pupils connect trigonometric ratios to solving problems on angles of elevation and depression.

Step 6: Worked Examples

Time: 8 minutes
Teaching Skill: Demonstration/Practice
Teacher’s Activity: The teacher works through the two examples presented in the board summary, explaining each step clearly and encouraging pupils to ask questions.
Pupils’ Activity: Pupils observe the teacher solving the examples, copy them into their notes, and ask questions where they need clarification.
Learning Point: Pupils learn how to apply the concepts and ratios to solve practical problems.

Step 7: Evaluation/Review

Time: 5 minutes
Teaching Skill: Questioning/Assessment
Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. Define an angle of elevation.
2. Define an angle of depression.
3. If an observer on the ground sees the top of a flagpole at an angle of elevation of 40°, what is the angle of depression of the observer from the top of the flagpole?
4. A person standing 20 meters away from the base of a tree measures the angle of elevation to the top of the tree as 45°. Calculate the height of the tree.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 2 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key points of the lesson, emphasizing the definitions of angles of elevation and depression and their application in solving real-world problems using trigonometry. The teacher assigns relevant homework.
Pupils’ Activity: Pupils listen to the summary and copy the assigned homework.
Learning Point: Learning is reinforced, and pupils are prepared for further practice.

Lesson Keywords

• Angle of Elevation – The angle formed when an observer looks upwards from a horizontal line.
• Angle of Depression – The angle formed when an observer looks downwards from a horizontal line.
• Line of Sight – The imaginary line from the observer’s eye to the object being viewed.
• Horizontal Line – A level line parallel to the ground.
• Trigonometric Ratios – Ratios of sides of a right-angled triangle (sine, cosine, tangent).

Differentiation

For pupils who are struggling, the teacher will provide simpler diagrams and more direct problems, focusing on identifying the angles and choosing the correct ratio. For advanced learners, the teacher will provide more complex problems involving two right-angled triangles or requiring multiple steps to solve.

Note for teachers using this lesson plan

Teachers should ensure pupils clearly understand the concept of the horizontal line. Emphasize the importance of drawing clear, labeled diagrams for each problem. Encourage pupils to use scientific calculators for trigonometric values. Practical demonstrations using real objects (like a tree and a measuring tape) can greatly enhance understanding.