Uganda NCDC Lesson Plans

The Unit Circle
1 / 20

S3 Mathematics - Term 1

Topic 1: Trigonometry I

Lesson 2: The Unit Circle

Duration: 40 minutes

Learning Outcomes

By the end of this lesson, you should be able to:

  • Understand what a unit circle is
  • Plot points on a unit circle for different angles
  • Relate coordinates to triangle sides

Review from Lesson 1

Key discovery:

  • Same angle = same ratio
  • Ratio depends only on the angle, not triangle size

Today's tool: The Unit Circle

  • Helps us visualize ALL possible angles

What is a Unit Circle?

Unit = 1

A circle with:

  • Radius = 1 unit
  • Center at origin (0, 0)

Why radius = 1? Makes calculations simple!

The Unit Circle

Unit circle with key points

Four Key Points

Angle Point Location
0 degrees (1, 0) Right
90 degrees (0, 1) Top
180 degrees (-1, 0) Left
270 degrees (0, -1) Bottom

Drawing Your Unit Circle

  1. Draw x and y axes on graph paper
  2. Mark origin (0, 0)
  3. Choose scale: 10 cm = 1 unit
  4. Use compass: draw circle with radius 10 cm
  5. Mark the four key points

Marking an Angle

To mark an angle (example: 30 degrees):

  1. Start from positive x-axis
  2. Measure counter-clockwise
  3. Draw line from origin to circle edge
  4. This point has coordinates (x, y)

Creating the Triangle

Unit circle creates right triangle

Measuring Coordinates

For each angle, we measure:

  • x-coordinate: horizontal distance
  • y-coordinate: vertical distance

Create a table:

Angle (degrees) x y
0 1.00 0.00
30 ? ?
45 ? ?
60 ? ?
90 0.00 1.00

Pair Activity

Measure and record coordinates for:

  • 30 degrees
  • 45 degrees
  • 60 degrees

Use your unit circle, protractor, and ruler!

Expected Values

Angle x (approx) y (approx)
30 degrees 0.87 0.50
45 degrees 0.71 0.71
60 degrees 0.50 0.87

Your measurements should be close!

Key Observation

  • At 0 degrees: all x, no y
  • At 90 degrees: no x, all y
  • In between: mix of both

Amazing fact: x squared + y squared = 1

(This is Pythagorean theorem!)

Connection to Triangles

Each point on the unit circle creates a right triangle:

  • Hypotenuse = 1 (radius)
  • x-coordinate = adjacent side
  • y-coordinate = opposite side

Preview: Trigonometric Functions

Soon we will learn:

  • y-coordinate will be called sine
  • x-coordinate will be called cosine
  • These are the famous trigonometric functions!

The Big Picture

The unit circle organizes all possible angles:

  • Each angle has unique x and y values
  • These values ARE the trigonometric ratios!
  • That's why the unit circle is so important

Exit Questions

  1. What is special about a unit circle?

  2. If you mark 45 degrees, approximately where will the point be?

  3. The x and y coordinates represent which parts of a triangle?

Homework

  1. Complete your table for angles: 0, 30, 45, 60, 90 degrees

  2. Add three more angles: 15, 75 degrees, and one of your choice

  3. For 60 degrees: find x and y, check if x squared + y squared = 1

Time: 20-25 minutes

Next Lesson

Lesson 3: Sine, Cosine, and Tangent Ratios

We will:

  • Define the three main trigonometric functions
  • Learn the formulas: SOH-CAH-TOA
  • Start solving triangle problems!

Credits

Created: December 2025
Based on: NCDC Lower Secondary Mathematics Syllabus (2019)

Source: National Curriculum Development Centre (NCDC), Uganda