Trigonometry

Grade 12 · Mathematics

Semester 1 | Period 2 | Week 11

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Subject: Mathematics

Semester: 1

Period: 2

Week: 11

WEEK 11

Class: Grade 12
Age: 17 years
Duration: 40 minutes for 5 periods
Subject: Mathematics
Topic: Trigonometry
Focus: Graph of Trigonometric Functions (Sine and Cosine)
Specific Objectives:
By the end of the lesson, students should be able to:

  1. Construct tables of values for sine and cosine for angles from 0° to 360°.
  2. Plot the sine and cosine graphs for angles 0° ≤ x ≤ 360°.
  3. Interpret the graphs of the sine and cosine functions.
  4. Identify key features of the sine and cosine graphs, such as amplitude, period, and phase shift.

Instructional Techniques:

  • Question and answer
  • Guided demonstration
  • Discussion
  • Practice exercises
  • Graphing exercises

Instructional Materials:

  • Graph board
  • Graph books
  • Pencil and ruler
  • Twine (for drawing axes)

PERIOD 1 & 2: Introduction to Trigonometric Graphs (Sine and Cosine)

Presentation:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the concept of trigonometric functions and their graphs. Explains the importance of sine and cosine functions.

Students listen attentively and ask questions.

Step 2 - Sine Function

Explains how to calculate sine values for various angles (0° ≤ x ≤ 360°). Discusses the key features of the sine graph (amplitude, period).

Students take notes on the key features and calculation of sine values.

Step 3 - Cosine Function

Explains the calculation of cosine values for various angles (0° ≤ x ≤ 360°) and how to plot these values on a graph. Discusses key features of the cosine graph.

Students take notes and ask questions about cosine values and plotting.

Step 4 - Constructing Tables

Guides students in constructing tables of values for sine and cosine, covering angles from 0° to 360°.

Students work in pairs to construct tables of values for sine and cosine.

Note on Board:
Sine Graph:

  • Amplitude: The maximum value of the sine function is 1, and the minimum is -1.
  • Period: The period is 360°, which means the function repeats every 360°.
    Cosine Graph:
  • Similar features to sine but starts from a maximum point at 0°.

Evaluation (5 Exercises):

  1. Write the sine values for the angles 0°, 30°, 60°, 90°, 180°, 270°, and 360°.
  2. Write the cosine values for the angles 0°, 30°, 60°, 90°, 180°, 270°, and 360°.
  3. What is the amplitude of the sine graph?
  4. What is the period of the cosine graph?
  5. How do the graphs of sine and cosine differ at x = 90°?

Classwork (5 Questions):

  1. Complete the table for sine values for 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°, and 360°.
  2. Complete the table for cosine values for 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°, and 360°.
  3. Sketch the sine graph for the range 0° ≤ x ≤ 360°.
  4. Sketch the cosine graph for the range 0° ≤ x ≤ 360°.
  5. Identify the key features of both sine and cosine graphs in the range 0° ≤ x ≤ 360°.

Assignment (5 Tasks):

  1. Write out the sine values for angles 0°, 15°, 30°, 45°, 60°, 75°, and 90°.
  2. Write out the cosine values for angles 0°, 15°, 30°, 45°, 60°, 75°, and 90°.
  3. Plot the graph of sine for the angles 0° to 360°.
  4. Plot the graph of cosine for the angles 0° to 360°.
  5. Discuss the practical applications of sine and cosine functions in real life.

 

PERIOD 3 & 4: Plotting Trigonometric Graphs

Presentation:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction to Graphing

Demonstrates how to plot the sine and cosine graphs using the values in the tables. Explains how to mark the axes and plot points.

Students follow along, drawing axes on their graph sheets.

Step 2 - Graphing Sine Function

Plots the sine graph for angles 0° ≤ x ≤ 360° on the graph board.

Students observe the graphing process and replicate it on their graph sheets.

Step 3 - Graphing Cosine Function

Plots the cosine graph for angles 0° ≤ x ≤ 360° on the graph board.

Students follow along, replicating the cosine graph on their own graph sheets.

Step 4 - Analyzing the Graphs

Discusses the key features of the graphs: amplitude, period, and phase shift.

Students analyze the graphs, noting key features and differences between sine and cosine.

Note on Board:

  • Sine Graph: Starts at 0°, goes up to +1 at 90°, back to 0 at 180°, down to -1 at 270°, and back to 0 at 360°.
  • Cosine Graph: Starts at +1 at 0°, goes down to 0 at 90°, down to -1 at 180°, up to 0 at 270°, and back to +1 at 360°.

Evaluation (5 Exercises):

  1. Plot the sine graph for angles from 0° to 360°.
  2. Plot the cosine graph for angles from 0° to 360°.
  3. Label the amplitude and period on the sine and cosine graphs.
  4. Identify the x-intercepts of the sine graph.
  5. Identify the maximum and minimum values of the cosine graph.

Classwork (5 Questions):

  1. Plot the sine graph for the angles 0°, 90°, 180°, 270°, and 360°.
  2. Plot the cosine graph for the angles 0°, 90°, 180°, 270°, and 360°.
  3. What is the phase shift of the sine and cosine graphs?
  4. How do you calculate the period of a trigonometric graph?
  5. Describe the symmetry of the sine and cosine graphs.

Assignment (5 Tasks):

  1. Plot the graph of sine for the angles 0° to 360°.
  2. Plot the graph of cosine for the angles 0° to 360°.
  3. Discuss how sine and cosine graphs are used in modeling periodic phenomena.
  4. Write out the steps to plot the sine graph from the table of values.
  5. Write out the steps to plot the cosine graph from the table of values.

 

PERIOD 5: Interpretation of Trigonometric Graphs

Presentation:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the concept of interpreting trigonometric graphs: reading key values such as amplitude, period, maximum, minimum, and intercepts.

Students listen attentively and ask clarifying questions.

Step 2 - Interpreting the Sine Graph

Guides students in reading values from the sine graph: maximum and minimum values, amplitude, and intercepts.

Students practice interpreting the sine graph.

Step 3 - Interpreting the Cosine Graph

Guides students in reading values from the cosine graph: maximum and minimum values, amplitude, and intercepts.

Students practice interpreting the cosine graph.

Step 4 - Practical Exercise

Provides several graphs for students to interpret. Emphasizes practical examples.

Students interpret the given graphs, identifying key features.

Note on Board:

  • Amplitude: The maximum value of the function.
  • Period: The length of one complete cycle.
  • Phase Shift: Horizontal displacement of the graph.
  • Intercepts: Points where the graph crosses the x-axis.

Evaluation (5 Exercises):

  1. What is the amplitude of the sine graph?
  2. What is the period of the cosine graph?
  3. Identify the x-intercepts of the sine graph.
  4. Identify the maximum value of the cosine graph.
  5. Describe the symmetry of the sine and cosine graphs.

Classwork (5 Questions):

  1. Interpret the given sine graph and identify its amplitude, period, and intercepts.
  2. Interpret the given cosine graph and identify its amplitude, period, and intercepts.
  3. Explain the phase shift in the sine graph.
  4. Explain the phase shift in the cosine graph.
  5. Compare and contrast the sine and cosine graphs.

Assignment (5 Tasks):

  1. Interpret the amplitude, period, and intercepts of the sine graph for 0° ≤ x ≤ 360°.
  2. Interpret the amplitude, period, and intercepts of the cosine graph for 0° ≤ x ≤ 360°.
  3. Discuss the practical application of sine and cosine graphs in real life (e.g., sound waves, light waves).
  4. Identify and interpret key features of a sine graph from a given set of values.
  5. Identify and interpret key features of a cosine graph from a given set of values.