Mathematics - Senior Secondary 1 - Trigonometry (III) – Trigonometric Ratios Related to the Unit Circle

Trigonometry (III) – Trigonometric Ratios Related to the Unit Circle

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  4. Trigonometry (III) – Trigonometric Ratios Related to the Unit Circle

Term: 3rd Term

Week: 7

Class: Senior Secondary School 1
Age: 15 years
Duration: 40 minutes of 5 periods
Subject: Mathematics
Topic: Trigonometry (III) – Trigonometric Ratios Related to the Unit Circle

 

SPECIFIC OBJECTIVES:

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

  1. Understand the relationship between trigonometric ratios and the unit circle.
  2. Construct a table of values for sine and cosine functions for 0° ≤ θ ≤ 360°.
  3. Plot the graphs of sine and cosine functions accurately.
  4. Use the graphs to estimate trigonometric values and solve problems.

 

INSTRUCTIONAL TECHNIQUES:

  • Explanation and illustration
  • Guided practice
  • Question and answer
  • Graph plotting activities
  • Peer discussion

 

INSTRUCTIONAL MATERIALS:

  • Graph boards
  • Graph books
  • Pencils and erasers
  • Mathematical sets
  • Protractors and rulers
  • Scientific calculators

 

PERIOD 1 & 2: Introduction to Unit Circle and Sine/Cosine Graphs

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the unit circle and explains how trigonometric ratios (sine and cosine) relate to angles and the circle. Demonstrates angle movement in the circle using a protractor.

Students listen and observe, noting the concept of angles and coordinate points.

Step 2 - Sine Values Table

Constructs a table of sine values for 0°, 30°, 45°, 60°, 90°... up to 360° using a calculator or geometry.

Students replicate the table in their books.

Step 3 - Cosine Values Table

Constructs a table of cosine values for the same angles.

Students write and compare the cosine values with sine values.

Step 4 - Coordinate Interpretation

Shows that points on the unit circle are (cos θ, sin θ) and relate directly to the tables.

Students visualize and match values with coordinates.

Step 5 - Preparation for Graph

Explains how to transfer these values onto a graph for plotting.

Students prepare their graph books.

 

NOTE ON BOARD:

  • Sine and cosine values are the y and x coordinates on the unit circle.
  • sin(θ) increases from 0 to 90°, decreases to 0 at 180°, then negative till 360°.
  • cos(θ) starts at 1, decreases to -1 and returns to 1 at 360°.

 

EVALUATION (5 Exercises):

  1. What is the sine of 90°?
  2. What is the cosine of 180°?
  3. Explain how the unit circle helps understand sine and cosine values.
  4. Complete the table for sin θ from 0° to 180°.
  5. Which coordinate does sine represent in the unit circle?

 

CLASSWORK (5 Questions):

  1. Draw the unit circle and label 0°, 90°, 180°, 270°, and 360°.
  2. Construct the table of values for sine from 0° to 360°.
  3. Construct the table of values for cosine from 0° to 360°.
  4. Identify when sine is 0, 1, and -1.
  5. Identify when cosine is 0, 1, and -1.

 

ASSIGNMENT (5 Tasks):

  1. Write out all values of sin θ and cos θ for every 30° from 0° to 360°.
  2. Research how the unit circle is applied in engineering.
  3. Illustrate the four quadrants of the unit circle and their signs.
  4. Compare the sine and cosine graphs by describing their wave shapes.
  5. Draw a sketch of the unit circle with one full rotation (0° to 360°).

 

PERIOD 3 & 4: Graph Plotting of Sine and Cosine Functions

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Sine Graph

Guides students to plot the sine graph using earlier values on a graph sheet. Points are plotted for every 30°.

Students follow along by plotting points and drawing the sine curve.

Step 2 - Cosine Graph

Similarly guides the plotting of cosine graph. Emphasizes periodic nature and wave shape.

Students plot cosine values on a new graph sheet.

Step 3 - Analysis

Helps students identify amplitude, period, and axis of both graphs.

Students discuss differences and similarities between the two graphs.

Step 4 - Real-Life Applications

Briefly discusses real-life uses like sound waves, tides, and alternating current.

Students ask questions and take notes.

 

NOTE ON BOARD:

  • Sine and cosine graphs are wave-like and periodic.
  • Sine starts from 0; cosine starts from 1.
  • Amplitude = 1, Period = 360°.
  • Important points to note: peak, trough, and zero crossings.

 

EVALUATION (5 Exercises):

  1. Plot the graph of sin θ from 0° to 360°.
  2. Plot the graph of cos θ from 0° to 360°.
  3. Identify maximum and minimum values of sin and cos.
  4. Describe the shape of each graph.
  5. State the period and amplitude of the sine and cosine graphs.

 

CLASSWORK (5 Questions):

  1. Draw the sine graph for θ = 0° to 360°.
  2. Draw the cosine graph for θ = 0° to 360°.
  3. At what angles does sin θ = 0?
  4. At what angles does cos θ = 1?
  5. What is the amplitude of both sine and cosine graphs?

 

ASSIGNMENT (5 Tasks):

  1. Draw both sine and cosine graphs on the same axis.
  2. Label all critical points (maxima, minima, intercepts).
  3. Research how sine and cosine are used in music and physics.
  4. Write a short note on the periodicity of trigonometric functions.
  5. Create a real-life scenario where a sine or cosine graph could be used.

 

PERIOD 5: Conclusion and Review

PRESENTATION:

  • Review sine and cosine definitions and their relation to the unit circle.
  • Discuss the behavior of the graphs and check for misconceptions.
  • Conduct a mini quiz to assess understanding.
  • Invite students to ask final questions.

 

EVALUATION:

  1. Quick check of graph plotting accuracy.
  2. Oral questioning based on sine and cosine values.
  3. Peer-to-peer review of plotted graphs and written work.

Assign points for graph interpretation and completeness.