Algebra II Lesson

The Unit Circle: From Right Triangles to Circular Motion

Shared by A STEM Educator

11th GradeBTC Style / Thinking Classrooms50 min

Learning Objective

Students will derive the coordinates of key points on the unit circle using special right triangle ratios, interpret those coordinates as (cos θ, sin θ), and evaluate trigonometric functions for angles in all four quadrants using reference angles.

Lesson Overview

Students build the unit circle from scratch using special right triangles, connect coordinates to cosine and sine, then apply reference angles across all four quadrants to model circular motion contexts like Ferris wheels and clock hands.

Materials

Vertical whiteboards or chart paper
Dry-erase markers (multiple colors if available)
Pencils and scratch paper
Basic calculator (for checking decimal approximations only)

Scaffolded Task Progression

1Task 1

A right triangle has a hypotenuse of length

1

and an angle of

45°

at the origin. The triangle sits in Quadrant I with the hypotenuse along the terminal side.

[ Interactive graph / diagram, available in the full lesson ]

Find the exact lengths of both legs. What are the coordinates of point

A

(the tip of the hypotenuse)?

2Task 2

Now use a

30°

60°

90°

triangle with hypotenuse

= 1

, with the

30°

angle at the origin in Quadrant I.

[ Interactive graph / diagram, available in the full lesson ]

Find the exact coordinates of point

A

(tip of the hypotenuse). Then repeat for a triangle with the

60°

angle at the origin — what are

A

's coordinates then?

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