Note for teachers using this lesson plan

Prepare the board, mathematical instruments and any calculator, logarithm table, graph sheet or model required for the worked examples. Teach graphs of sine and cosine functions with correct notation and orderly working, checking each stage before students proceed. By the end, students should be able to explain the method and solve representative problems accurately.

Class: Senior Secondary School 3 (SS3 / SSS3)
Term: First Term
Week: 9 – 10
Age: 17 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Trigonometry
Previous Lesson:
Topic: TRIGONOMETRY
Subject Matter: Graph of trigonometric functions sine and cosine for 0≤x≤360°, interpretation of graphs of trigonometric functions

Specific Objectives

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

Cognitive Domain:

Construct tables of values for sine and cosine functions for 0° ≤ x ≤ 360°.
Plot the graphs of y = sin x and y = cos x for 0° ≤ x ≤ 360°.
Interpret the key features of the graphs of y = sin x and y = cos x.
Read values from the graphs of trigonometric functions.

Affective Domain:

Appreciate the visual representation of trigonometric functions.
Develop accuracy and neatness in drawing graphs.

Psychomotor Domain:

Use graph paper, ruler, and pencil effectively to draw smooth curves.
Accurately plot points to represent trigonometric functions.

Collaborate with peers when constructing tables and plotting graphs.

Reference Materials

The following resources were used in planning this lesson:

FCT-ERC SS 3 Scheme of Work, September 2014 curriculum.
Relevant State Unified Scheme of Work.
A suitable General Mathematics textbook for SS 3.

Instructional Materials

The teacher will teach this lesson with the aid of:

Graph board
Graph books
Ruler
Pencil

Rationale for the Lesson

This lesson helps students understand how trigonometric functions behave visually, which is important for solving problems involving periodic phenomena like sound waves or alternating current. Understanding these graphs enables students to visualize mathematical relationships and interpret real-world data more effectively.

Prerequisite/Previous Knowledge

Students have prior knowledge of basic trigonometric ratios, angles in a circle, coordinate geometry, and plotting points on a graph.

Lesson Content/Board Summary

Graph of Trigonometric Functions

Introduction to Trigonometric Graphs

Trigonometric graphs are visual representations of trigonometric functions such as sine, cosine, and tangent. These graphs are periodic, meaning their patterns repeat over regular intervals. Understanding these graphs helps in analyzing oscillatory motions and waves.

Graph of y = sin x for 0° ≤ x ≤ 360°

To plot the graph of y = sin x, a table of values is first constructed, then the points are plotted on a Cartesian plane and joined by a smooth curve.

Steps for Construction and Plotting:

Construct a Table of Values: Choose various values for x (angles) within the given range (0° to 360°), usually at intervals of 30° or 45°, and find the corresponding values of sin x.
Example Table for y = sin x:

x 0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360°

sin x (approx.) 0 0.5 0.87 1 0.87 0.5 0 -0.5 -0.87 -1 -0.87 -0.5 0
Draw Axes: Draw a horizontal x-axis (for angles) and a vertical y-axis (for sin x values).
Choose Scale: Select appropriate scales for both axes. For example, 1 cm to 30° on the x-axis and 1 cm to 0.2 units on the y-axis.
Plot Points: Plot the (x, sin x) coordinate pairs from the table.
Draw Smooth Curve: Join the plotted points with a freehand smooth curve. Do not use a ruler for drawing the curve.

The following are key features of the graph of y = sin x:

Amplitude: The maximum displacement from the x-axis, which is 1.
Period: The interval over which the graph completes one full cycle and begins to repeat, which is 360° (or 2π radians).
Maximum Value: The highest value of y, which is 1 (occurs at x = 90°).
Minimum Value: The lowest value of y, which is -1 (occurs at x = 270°).
x-intercepts: The points where the graph crosses the x-axis (y=0), which are at 0°, 180°, and 360°.
y-intercept: The point where the graph crosses the y-axis (x=0), which is at y = 0.

Graph of y = cos x for 0° ≤ x ≤ 360°

Similar to the sine graph, the graph of y = cos x is constructed by first generating a table of values and then plotting these points.

Steps for Construction and Plotting:

Construct a Table of Values: Choose various values for x (angles) within the given range (0° to 360°), and find the corresponding values of cos x.
Example Table for y = cos x:

x 0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360°

cos x (approx.) 1 0.87 0.5 0 -0.5 -0.87 -1 -0.87 -0.5 0 0.5 0.87 1
Draw Axes: Draw a horizontal x-axis (for angles) and a vertical y-axis (for cos x values).
Choose Scale: Select appropriate scales for both axes.
Plot Points: Plot the (x, cos x) coordinate pairs from the table.
Draw Smooth Curve: Join the plotted points with a freehand smooth curve.

The following are key features of the graph of y = cos x:

Amplitude: The maximum displacement from the x-axis, which is 1.
Period: The interval over which the graph completes one full cycle and begins to repeat, which is 360° (or 2π radians).
Maximum Value: The highest value of y, which is 1 (occurs at x = 0° and 360°).
Minimum Value: The lowest value of y, which is -1 (occurs at x = 180°).
x-intercepts: The points where the graph crosses the x-axis (y=0), which are at 90° and 270°.
y-intercept: The point where the graph crosses the y-axis (x=0), which is at y = 1.

Interpretation of Graphs of Trigonometric Functions

Interpreting trigonometric graphs involves understanding what different parts of the graph represent.

The following are aspects to interpret from trigonometric graphs:

Reading Values: For any given angle (x-value) on the x-axis, find the corresponding sine or cosine value (y-value) by moving vertically to the curve and then horizontally to the y-axis. Conversely, for a given sine or cosine value, find the corresponding angle(s).
Amplitude: Half the difference between the maximum and minimum values of the function. It indicates the “height” of the wave.
Period: The horizontal length of one complete cycle of the wave. It shows how often the pattern repeats.
Maximum and Minimum Values: The highest and lowest points reached by the graph, respectively.
x-intercepts: The angles where the function’s value is zero.
y-intercept: The value of the function when the angle is zero.

Additional Graph Examples

From the standard sine values, $\sin 30^\circ=0.5$ and $\sin 150^\circ=0.5$ , so $$y=0.5$$ meets the sine curve at two points between $0^\circ$ and $360^\circ$ .
For the cosine graph, $\cos 60^\circ=0.5$ and $\cos 300^\circ=0.5$ , giving two solutions to $\cos x=0.5$ in one complete cycle.

Teaching Methods/Instructional Techniques

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

Instructional Procedures

Step 1: Introduction and Recall

Time: 4 minutes

Teaching Skill: Set Induction

Teacher’s Activity: The teacher briefly reviews previous knowledge of trigonometric ratios and coordinate geometry. The teacher then introduces the concept of representing these ratios graphically, explaining that today's lesson will focus on plotting and interpreting the graphs of y = sin x and y = cos x.

Students’ Activity: Students listen attentively and recall their previous knowledge about trigonometry and graphs.

Learning Point: Prior trigonometry knowledge.

Step 2: Construction of Table of Values for y = sin x

Time: 5 minutes

Teaching Skill: Explanation/Guidance

Teacher’s Activity: The teacher guides students on how to construct a table of values for y = sin x for angles from 0° to 360° at 30° intervals. The teacher demonstrates how to find sine values for key angles and encourages students to use their calculators for other values.

Students’ Activity: Students actively participate in constructing the table of values in their graph books, calculating sine values for different angles.