Lesson 4.1

Unrolling the Unit Circle

What happens when you take the -coordinates from the unit circle and stretch them out along a horizontal axis? You get the most important curve in mathematics: the sine wave.

Introduction

Until now, the unit circle has been our tool for finding exact trig values at specific angles. But what does the overall behaviour of sine look like as the angle sweeps from through and beyond? The answer: a smooth, repeating wave.

Past Knowledge

On the unit circle, is the -coordinate of the point at angle .

Today's Goal

Plot and identify its key features: period, amplitude, midline, and zeros.

Future Success

Every transformation in the next several lessons modifies this baseline sine wave.

Key Concepts

From Circle to Wave

Imagine a point traveling counterclockwise around the unit circle starting at . At each position, record the -coordinate (which is ) and plot it against the angle on a standard -plane. The result is the sine curve.

The Graph of

Key Features of

Feature	Value	Meaning
Amplitude		Maximum distance from the midline
Period		Length of one full cycle
Midline		Horizontal axis of symmetry
Maximum	at	Crest of the wave
Minimum	at	Trough of the wave
Zeros		Where the wave crosses the midline

The Quarter-Period Pattern

Every sine wave follows the same sequence within one period: zero → max → zero → min → zero. These five key points are evenly spaced, each separated by .

Worked Examples

Basic

Reading the Sine Graph

Question: From the graph of , what is the value of ?

Step 1: Locate on the horizontal axis (quarter of the way through one period).

Step 2: At this point, the sine wave is at its maximum height: .

Final Answer:

Intermediate

Identifying the Five Key Points

Question: List the five key points of for one complete cycle starting at .

Step 1: The period is , so the quarter-period is .

Step 2: Start at and add four times:

Step 3: Apply the pattern zero → max → zero → min → zero:

Final Answer: The five key points are , , , , .

Advanced

Determining Behavior from the Graph

Question: On what interval(s) is negative during one full period?

Step 1: The sine wave crosses zero at (going from positive to negative) and at (returning to zero).

Step 2: Between and , the wave is below the -axis.

Final Answer: on the interval .

Common Pitfalls

Using Degrees on the X-Axis

When graphing , the -axis is in radians, not degrees. The period is , not . If your graph extends to , you are graphing complete cycles!

Real-Life Applications

Sound Waves

Every pure musical note is a sine wave. A tuning fork vibrating at 440 Hz (the note A) produces a pressure wave described by . The amplitude determines loudness, the period determines pitch, and adding multiple sine waves together (via Fourier analysis) recreates any sound — from a violin to the human voice.