Class: Senior Secondary School 2 (SS2 / SSS2)
Term: 3rd Term
Week: 2
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Trigonometry
Previous Lesson: Trigonometric Function: Meaning, Domain, Range and Graphs.
Topic: TRIGONOMETRIC FUNCTION
Subject Matter: Relationship between graphs of trigonometric ratios (sin x and sin 2x, y = a sin(bx) + c, y = a cos(bx) + c, y = a tan(bx) + c), graphs of inverse trig ratios

Specific Objectives

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

Cognitive Domain:

Identify the relationship between the graphs of trigonometric ratios like sin x and sin 2x.
Explain the effects of amplitude, period, and vertical shifts on trigonometric graphs of the form y = a sin(bx) + c, y = a cos(bx) + c, and y = a tan(bx) + c.
State the principal value ranges for inverse trigonometric functions.

Affective Domain:

Appreciate the importance of understanding transformations in trigonometric graphs.
Show interest in sketching and analyzing various trigonometric and inverse trigonometric functions.

Psychomotor Domain:

Sketch graphs of transformed trigonometric functions accurately.
Sketch graphs of inverse trigonometric functions (arcsin x, arccos x, arctan x).

Collaborate with peers to discuss and solve problems related to trigonometric graphs.

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 2
https://www.mathsisfun.com/algebra/trig-inverse.html
https://www.khanacademy.org/math/algebra2/trig-functions/trig-transformations/v/amplitude-and-period

Instructional Materials

The teacher will teach this lesson with the aid of:

Charts showing sketches of inverse of sin x, cos x and tan x
Graph boards or grid paper
Markers/chalk

Rationale for the Lesson

This lesson helps pupils understand how changes in parameters affect the shape and position of trigonometric graphs. This knowledge is important for solving real-world problems involving periodic phenomena and for advanced studies in mathematics and science.

Prerequisite/Previous Knowledge

Pupils should have prior knowledge of basic trigonometric ratios, plotting points on a graph, and sketching the fundamental graphs of y = sin x, y = cos x, and y = tan x.

Lesson Content/Board Summary

Trigonometric Functions and their Graphs

Relationship between graphs of trigonometric ratios (Transformations)

The general forms of transformed trigonometric graphs are:

For sine and cosine: y = a sin(bx) + c or y = a cos(bx) + c
For tangent: y = a tan(bx) + c

Where:

a: Represents the amplitude (for sin/cos) or vertical stretch/compression (for tan). The amplitude is |a|.
b: Affects the period of the graph.
- For sin/cos functions, the period is T = 2π/|b| (or 360°/|b|).
- For tan functions, the period is T = π/|b| (or 180°/|b|).
c: Represents the vertical shift of the graph. A positive ‘c’ shifts the graph upwards, and a negative ‘c’ shifts it downwards.

Worked Examples:

Example 1: Compare the graphs of y = sin x and y = sin 2x for 0 ≤ x ≤ 2π.

Step 1: Identify parameters.

For y = sin x: a = 1, b = 1, c = 0. Amplitude = 1, Period = 2π/1 = 2π.
For y = sin 2x: a = 1, b = 2, c = 0. Amplitude = 1, Period = 2π/2 = π.

Step 2: Compare.

Both graphs have the same amplitude (1).
The graph of y = sin 2x has half the period of y = sin x, meaning it completes two full cycles in the same interval where y = sin x completes one cycle.

Example 2: Sketch the graph of y = 2 cos x + 1 for 0 ≤ x ≤ 2π.

Step 1: Identify parameters.

a = 2 (Amplitude = 2)
b = 1 (Period = 2π/1 = 2π)
c = 1 (Vertical shift = 1 unit upwards)

Step 2: Determine key points.

The standard y = cos x graph oscillates between -1 and 1.
With amplitude 2, it oscillates between -2 and 2.
With a vertical shift of +1, the graph oscillates between -2+1 = -1 and 2+1 = 3. The midline is y = 1.

Step 3: Sketch the graph.

Plot points for y = 2 cos x + 1 at x = 0, π/2, π, 3π/2, 2π.
y(0) = 2(1) + 1 = 3
y(π/2) = 2(0) + 1 = 1
y(π) = 2(-1) + 1 = -1
y(3π/2) = 2(0) + 1 = 1
y(2π) = 2(1) + 1 = 3
Connect the points with a smooth curve.

Graphs of Inverse Trigonometric Ratios

Inverse trigonometric functions are used to find the angle when the value of the trigonometric ratio is known. For these inverses to be functions, the domain of the original trigonometric function must be restricted to ensure it is one-to-one.

The main inverse trigonometric functions are:

y = arcsin x (or sin⁻¹ x): This means ‘the angle whose sine is x’.
- Domain: [-1, 1]
- Range (Principal Value): [-π/2, π/2] or [-90°, 90°]
y = arccos x (or cos⁻¹ x): This means ‘the angle whose cosine is x’.
- Domain: [-1, 1]
- Range (Principal Value): [0, π] or [0°, 180°]
y = arctan x (or tan⁻¹ x): This means ‘the angle whose tangent is x’.
- Domain: (-∞, ∞)
- Range (Principal Value): (-π/2, π/2) or (-90°, 90°)

Note: The graph of an inverse function can be obtained by reflecting the graph of the original function (with its restricted domain) across the line y = x.

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 the basic graphs of y = sin x, y = cos x, and y = tan x and asks pupils what happens if we multiply or add numbers to these functions.
Pupils’ Activity: Pupils recall the basic graphs and share their ideas.
Learning Point: Pupils activate prior knowledge and are introduced to the concept of graph transformations.

Step 2: Exploring Amplitude and Period

Time: 8 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher explains how the ‘a’ and ‘b’ parameters in y = a sin(bx) + c affect the amplitude and period of the graph, using examples like sin x vs 2 sin x and sin x vs sin 2x.
Pupils’ Activity: Pupils observe the explanations, ask questions, and note the definitions of amplitude and period.
Learning Point: Pupils understand how ‘a’ changes the height and ‘b’ changes the length of a cycle.

Step 3: Exploring Vertical Shifts

Time: 8 minutes
Teaching Skill: Explanation
Teacher’s Activity: The teacher explains how the ‘c’ parameter in y = a sin(bx) + c causes a vertical shift (up or down) of the graph, demonstrating with an example like y = sin x + 1.
Pupils’ Activity: Pupils follow the explanation and understand the concept of vertical translation.
Learning Point: Pupils learn how ‘c’ moves the entire graph up or down.

Step 4: Sketching Transformed Graphs

Time: 8 minutes
Teaching Skill: Guided Practice
Teacher’s Activity: The teacher guides pupils to sketch a transformed trigonometric graph, such as y = 2 cos x + 1, on their graph paper, emphasizing the steps involved.
Pupils’ Activity: Pupils actively participate in sketching the transformed graph, applying the concepts learned.
Learning Point: Pupils develop practical skills in drawing transformed trigonometric graphs.

Step 5: Introduction to Inverse Trigonometric Functions

Time: 8 minutes
Teaching Skill: Explanation/Questioning
Teacher’s Activity: The teacher introduces the concept of inverse trigonometric functions (arcsin x, arccos x, arctan x), explaining why the domain of the original function must be restricted. The teacher also states their principal value ranges.
Pupils’ Activity: Pupils listen attentively, ask clarifying questions, and copy the definitions and ranges.
Learning Point: Pupils understand the purpose and necessary restrictions for inverse trigonometric functions.

Step 6: Sketching Inverse Trigonometric Graphs

Time: 8 minutes
Teaching Skill: Demonstration
Teacher’s Activity: The teacher demonstrates how to sketch the graphs of inverse trigonometric functions (e.g., y = arcsin x) by reflecting the restricted original graph across the line y = x, using charts as aids.
Pupils’ Activity: Pupils observe the demonstration and understand the graphical representation of inverse functions.
Learning Point: Pupils grasp the visual representation and sketching technique for inverse trigonometric graphs.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

Explain the effect of ‘a’ on the graph of y = a sin(bx) + c.
What is the period of the graph y = cos(3x)?
State the principal value range for y = arcsin x.
Describe how the graph of y = tan x + 2 differs from 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 about graph transformations and inverse trigonometric functions. The teacher also assigns relevant homework from the textbook.
Pupils’ Activity: Pupils listen to the summary and copy down the homework assignment.
Learning Point: Pupils consolidate their understanding of the lesson and prepare for independent practice.

Lesson Keywords

Amplitude – The maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position.
Period – The length of one complete cycle of a repeating waveform.
Vertical Shift – The vertical translation of a graph, moving it up or down.
Inverse Trigonometric Functions – Functions that give the angle corresponding to a given trigonometric ratio.
Principal Value Range – The restricted range of an inverse trigonometric function that ensures it is a single-valued function.

Differentiation

The teacher will provide additional support for struggling learners by giving simplified examples and one-on-one guidance. Advanced learners will be challenged with more complex transformation problems and encouraged to explore phase shifts (horizontal shifts) independently.

Note for teachers using this lesson plan

Ensure that pupils have access to graph paper for sketching. Emphasize the connection between the parameters and the visual changes in the graphs. When introducing inverse functions, clearly explain the need for domain restriction and the concept of principal values. Encourage pupils to use different colours when sketching multiple graphs on the same axis for clarity.