Class: Senior Secondary School 1 (SS1, SS 1, SSS1, SSS 1)
Term: 3rd Term
Week: 6
Age: 15 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: General Mathematics
Previous Lesson: Trigonometry.
Topic: TRIGONOMETRY (III)
Subject Matter: Trigonometric ratios related to the unit circle; Drawing graphs of sine and cosine for 0° ≤ θ ≤ 360°; Constructing tables of values and obtaining solutions from graphs.

Specific Objectives

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

Cognitive Domain:

Define the unit circle and relate trigonometric ratios to it.
Construct tables of values for sine and cosine functions for angles from 0° to 360°.
Interpret information presented on sine and cosine graphs.

Affective Domain:

Appreciate the importance of graphical representation of trigonometric functions.
Show interest in drawing accurate trigonometric graphs.

Psychomotor Domain:

Draw the graphs of y = sin θ and y = cos θ for 0° ≤ θ ≤ 360°.
Obtain solutions to trigonometric equations by reading values from graphs.

Participate cooperatively in group activities related to graph construction.

Reference Materials

The following resources were used in planning this lesson:

9 Years Basic Education Curriculum for Mathematics (Senior Secondary Education)
State Unified Scheme of Work for Senior Secondary School Mathematics
New General Mathematics for Senior Secondary Schools 1
https://www.mathsisfun.com/geometry/unit-circle.html
https://www.youtube.com/watch?v=s_KjE_L641s

Instructional Materials

The teacher will teach this lesson with the aid of:

Graph board
Graph book
Pencils
Mathematical sets (ruler, protractor, compass)

Rationale for the Lesson

This lesson helps pupils understand how trigonometric ratios are derived from the unit circle and how they can be represented graphically. This understanding is important for solving real-world problems involving oscillating phenomena like sound waves, light waves, and electrical currents, making mathematics relevant to daily life.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of basic trigonometric ratios (SOH CAH TOA), angles, coordinate geometry, and plotting points on a graph.

Lesson Content/Board Summary

Trigonometric Ratios and Graphs of Sine and Cosine

1. The Unit Circle and Trigonometric Ratios

A unit circle is a circle with a radius of one unit, centered at the origin (0,0) of a coordinate plane. For any angle θ measured counter-clockwise from the positive x-axis, the coordinates (x, y) of the point where the terminal side of the angle intersects the unit circle represent the cosine and sine of the angle, respectively.

Therefore:

cos θ = x-coordinate
sin θ = y-coordinate

2. Graph of y = sin θ for 0° ≤ θ ≤ 360°

To draw the graph of y = sin θ, a table of values is constructed by calculating the sine of various angles. The angles are plotted on the horizontal axis (x-axis) and the corresponding sine values on the vertical axis (y-axis).

Steps to draw the graph of y = sin θ:

Construct a table of values for sin θ for θ from 0° to 360° at suitable intervals (e.g., 30°, 45°, 90°).
Choose appropriate scales for the x-axis (angles) and y-axis (sine values).
Plot the points from the table on the graph paper.
Draw a smooth curve connecting the plotted points.

3. Graph of y = cos θ for 0° ≤ θ ≤ 360°

Similar to the sine graph, the cosine graph is drawn by plotting angles against their corresponding cosine values. The shape of the cosine graph is identical to the sine graph but shifted horizontally.

Steps to draw the graph of y = cos θ:

Construct a table of values for cos θ for θ from 0° to 360° at suitable intervals.
Choose appropriate scales for the x-axis (angles) and y-axis (cosine values).
Plot the points from the table on the graph paper.
Draw a smooth curve connecting the plotted points.

4. Obtaining Solutions from Graphs

Trigonometric graphs can be used to find approximate solutions to trigonometric equations or to find values of sine or cosine for specific angles.

Ways to obtain solutions from graphs:

To find sin θ for a given θ: Locate θ on the x-axis, move vertically to the curve, then horizontally to the y-axis to read the sine value.
To solve sin θ = k (where k is a value): Locate k on the y-axis, move horizontally to intersect the curve, then vertically down to the x-axis to read the angle(s).

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 greets the pupils and reviews the previous lesson on basic trigonometric ratios. The teacher then asks pupils to recall the maximum and minimum values of sine and cosine.
Pupils’ Activity: Pupils respond to questions and recall previous knowledge.
Learning Point: Pupils connect the new lesson to prior knowledge and are prepared for the topic.

Step 2: Explanation of Unit Circle and Trigonometric Ratios

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher draws a unit circle on the board and explains how the x and y coordinates of a point on the circle relate to the cosine and sine of the angle formed with the positive x-axis. The teacher demonstrates with a few key angles (0°, 90°, 180°, 270°, 360°).
Pupils’ Activity: Pupils observe the demonstration, ask questions, and take notes.
Learning Point: Pupils understand the definition of a unit circle and its relationship to sine and cosine.

Step 3: Constructing Table of Values and Drawing Graph of y = sin θ

Time: 10 minutes
Teaching Skill: Guided Practice/Demonstration
Teacher’s Activity: The teacher guides pupils to construct a table of values for y = sin θ for 0° ≤ θ ≤ 360° using intervals of 30° or 45°. The teacher then demonstrates on the graph board how to plot these points and draw a smooth curve for the sine graph.
Pupils’ Activity: Pupils participate in calculating values, construct their tables in their graph books, and begin plotting points under guidance.
Learning Point: Pupils learn how to prepare data and begin drawing the sine graph.

Step 4: Constructing Table of Values and Drawing Graph of y = cos θ

Time: 8 minutes
Teaching Skill: Guided Practice/Demonstration
Teacher’s Activity: The teacher guides pupils to construct a table of values for y = cos θ for 0° ≤ θ ≤ 360° using similar intervals. The teacher demonstrates how to plot these points and draw a smooth curve for the cosine graph, highlighting its relationship to the sine graph.
Pupils’ Activity: Pupils construct their tables for cosine and draw the cosine graph in their graph books.
Learning Point: Pupils learn to draw the cosine graph and observe its characteristics.

Step 5: Obtaining Solutions from Graphs

Time: 5 minutes
Teaching Skill: Application/Problem Solving
Teacher’s Activity: The teacher demonstrates how to use the drawn graphs to obtain values (e.g., find sin 150°) and solve simple trigonometric equations (e.g., find θ when cos θ = 0.5).
Pupils’ Activity: Pupils follow the demonstration and practice reading values or solutions from their graphs.
Learning Point: Pupils learn the practical application of trigonometric graphs in solving problems.

Step 6: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

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

Define a unit circle.
State the relationship between the coordinates on a unit circle and sine/cosine values.
List two steps involved in drawing the graph of y = sin θ.
Explain how you would use the graph of y = cos θ to find the value of θ when cos θ = -0.8.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 7: Conclusion

Time: 2 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key learning points, emphasizing the importance of accurate graph plotting and interpretation. The teacher gives an assignment: “Draw the graphs of y = sin θ and y = cos θ on the same axes for 0° ≤ θ ≤ 360° and use them to find the values of sin 210° and cos 300°.”
Pupils’ Activity: Pupils listen attentively and copy the assignment.
Learning Point: Pupils consolidate their learning and apply it in a take-home task.

Lesson Keywords

Unit Circle – A circle with a radius of one unit, centered at the origin of a coordinate plane.
Sine Graph – The graphical representation of the function y = sin θ.
Cosine Graph – The graphical representation of the function y = cos θ.
Table of Values – A list of corresponding x and y values used to plot a graph.
Trigonometric Ratios – Ratios of the sides of a right-angled triangle, such as sine, cosine, and tangent.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide additional challenges, such as sketching graphs for different amplitudes or periods. For those who need more support, the teacher will provide more one-on-one guidance during graph construction and may simplify the range of angles or provide partially completed tables of values.

Note for teachers using this lesson plan

Ensure that pupils have access to graph paper and mathematical sets. Emphasize the importance of clear labeling of axes and accurate plotting for effective graph interpretation. Encourage pupils to use different colours for sine and cosine graphs when drawn on the same axes.