Class: Senior Secondary School 2 (SS2 / SSS2)
Term: Third Term
Week: 1
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Trigonometry
Previous Lesson: Replacement Model: Meaning and Replacement Policies.
Topic: TRIGONOMETRIC FUNCTION
Subject Matter: Meaning of trigonometric function, knowledge of six trigonometric functions, range and domain of specified trigonometric functions, graphs of trigonometric ratios (amplitude, periodicity).

Specific Objectives

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

Cognitive Domain:

Define trigonometric functions.
Identify the six trigonometric functions.
State the domain and range for sine, cosine, and tangent functions.

Affective Domain:

Appreciate the importance of trigonometric functions in various mathematical and scientific applications.
Show interest in learning how to sketch trigonometric graphs.

Psychomotor Domain:

Sketch the basic graphs of sine, cosine, and tangent functions.
Accurately label the amplitude and period on trigonometric graphs.

Collaborate with peers in identifying characteristics of trigonometric functions.

Reference Materials

The following resources were used in planning this lesson:

Senior Secondary Schools Education Curriculum
State Unified Scheme of Work
New Further Mathematics for Senior Secondary Schools by P.N. Okeke

Instructional Materials

The teacher will teach this lesson with the aid of:

Charts showing the relationship between the six trigonometric ratios.
Whiteboard and markers.
Graph papers.

Rationale for the Lesson

This lesson helps pupils understand the fundamental relationships between angles and sides of triangles, which is essential for advanced mathematics and real-world applications in engineering, physics, and navigation. It builds a foundation for solving complex problems involving periodic phenomena.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of basic trigonometry (SOH CAH TOA), angles, and coordinate geometry.

Lesson Content/Board Summary

Trigonometric Functions

Meaning of Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a right-angled triangle to the ratios of the lengths of its sides. They are also defined using the coordinates of points on a unit circle.

The Six Trigonometric Functions

The six basic trigonometric functions are:

Sine (sin): Opposite side / Hypotenuse (y/r on unit circle)
Cosine (cos): Adjacent side / Hypotenuse (x/r on unit circle)
Tangent (tan): Opposite side / Adjacent side (y/x on unit circle)
Cosecant (csc): Reciprocal of sine (1/sin) = Hypotenuse / Opposite side (r/y)
Secant (sec): Reciprocal of cosine (1/cos) = Hypotenuse / Adjacent side (r/x)
Cotangent (cot): Reciprocal of tangent (1/tan) = Adjacent side / Opposite side (x/y)

Domain and Range of Trigonometric Functions

The domain is the set of all possible input values (angles), and the range is the set of all possible output values (ratios).

For y = sin x:
- Domain: All real numbers (–∞ < x < ∞) or (0° to 360° for one cycle).
- Range: -1 ≤ y ≤ 1.
For y = cos x:
- Domain: All real numbers (–∞ < x < ∞) or (0° to 360° for one cycle).
- Range: -1 ≤ y ≤ 1.
For y = tan x:
- Domain: All real numbers except x = (n + 1/2)π, where n is an integer (i.e., odd multiples of 90°).
- Range: All real numbers (–∞ < y < ∞).

Graphs of Trigonometric Ratios

The graphs of trigonometric functions are periodic, meaning their patterns repeat at regular intervals.

Amplitude: The maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position. For y = A sin(Bx) or y = A cos(Bx), the amplitude is |A|.
Period: The length of one complete cycle of the wave. For y = sin(Bx) or y = cos(Bx), the period is 360°/|B| or 2π/|B| radians. For y = tan(Bx), the period is 180°/|B| or π/|B| radians.

To sketch y = sin x:

Step 1: Identify amplitude (1) and period (360° or 2π).
Step 2: Plot key points: (0,0), (90,1), (180,0), (270,-1), (360,0).
Step 3: Draw a smooth curve through these points.

To sketch y = cos x:

Step 1: Identify amplitude (1) and period (360° or 2π).
Step 2: Plot key points: (0,1), (90,0), (180,-1), (270,0), (360,1).
Step 3: Draw a smooth curve through these points.

To sketch y = tan x:

Step 1: Identify period (180° or π). No amplitude.
Step 2: Note vertical asymptotes at x = 90°, 270°, etc.
Step 3: Plot key points: (0,0), (45,1), (-45,-1) within each period.
Step 4: Draw a curve approaching the asymptotes.

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 reviews previous knowledge on basic trigonometry (SOH CAH TOA) and asks pupils to recall what they know about angles and ratios in a right-angled triangle.
Pupils’ Activity: Pupils respond to the teacher’s questions and share their prior knowledge of trigonometry.
Learning Point: Pupils recall basic trigonometric concepts.

Step 2: Meaning of Trigonometric Functions

Time: 7 minutes
Teaching Skill: Explanation
Teacher’s Activity: The teacher explains the meaning of trigonometric functions, defining them as functions that relate angles to ratios of sides in triangles and on a unit circle.
Pupils’ Activity: Pupils listen attentively and ask questions for clarification.
Learning Point: Pupils understand the fundamental definition of trigonometric functions.

Step 3: The Six Trigonometric Functions

Time: 7 minutes
Teaching Skill: Listing/Identification
Teacher’s Activity: The teacher introduces the six trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) and explains their definitions using both right-angled triangles and the unit circle concept.
Pupils’ Activity: Pupils identify the six functions and note down their definitions.
Learning Point: Pupils know the names and definitions of the six trigonometric functions.

Step 4: Domain and Range of Trigonometric Functions

Time: 7 minutes
Teaching Skill: Guided Discovery
Teacher’s Activity: The teacher guides pupils to determine the domain and range for sine, cosine, and tangent functions, explaining why certain values are excluded for tangent.
Pupils’ Activity: Pupils state the domain and range for the specified trigonometric functions.
Learning Point: Pupils understand the permissible input and output values for sine, cosine, and tangent.

Step 5: Graphs of Sine and Cosine

Time: 7 minutes
Teaching Skill: Demonstration/Illustration
Teacher’s Activity: The teacher demonstrates how to sketch the graphs of y = sin x and y = cos x, emphasizing their amplitude and periodicity on the whiteboard.
Pupils’ Activity: Pupils observe the sketching process and take notes on the key features of the graphs.
Learning Point: Pupils learn how to sketch sine and cosine graphs and identify their amplitude and period.

Step 6: Graph of Tangent

Time: 5 minutes
Teaching Skill: Demonstration
Teacher’s Activity: The teacher demonstrates how to sketch the graph of y = tan x, highlighting its periodic nature and the presence of asymptotes.
Pupils’ Activity: Pupils observe the sketching and understand the unique characteristics of the tangent graph.
Learning Point: Pupils learn how to sketch the tangent graph, noting its period and asymptotes.

Step 7: Evaluation/Review

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

Define a trigonometric function.
List the six trigonometric functions.
State the domain and range of y = sin x.
What is the period of y = tan x?

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 learning points of the lesson and assigns homework, which includes sketching graphs of trigonometric functions.
Pupils’ Activity: Pupils listen to the summary and copy down the homework assignment.
Learning Point: Pupils reinforce their understanding and prepare for independent practice.

Lesson Keywords

Trigonometric function – A function relating an angle of a right triangle to the ratio of two sides.
Sine – Opposite side / Hypotenuse.
Cosine – Adjacent side / Hypotenuse.
Tangent – Opposite side / Adjacent side.
Secant – Reciprocal of cosine.
Cosecant – Reciprocal of sine.
Cotangent – Reciprocal of tangent.
Domain – The set of all possible input values for a function.
Range – The set of all possible output values for a function.
Amplitude – The maximum displacement of a periodic wave from its equilibrium position.
Period – The length of one complete cycle of a periodic function.

Differentiation

For pupils who grasp concepts quickly, the teacher can introduce transformations of trigonometric graphs (e.g., y = A sin(Bx + C) + D). For those needing more support, the teacher will provide additional guided practice on identifying the six ratios and plotting basic points for graphs.

Note for teachers using this lesson plan

Ensure pupils have access to graph paper for sketching activities. Emphasize the connection between the unit circle and the graphs of trigonometric functions. Encourage active participation and questioning throughout the lesson.