Period
Period controls how “wide” each cycle is. A larger value compresses the wave horizontally, creating faster oscillations — a smaller stretches it out.
Introduction
The standard sine wave completes one full cycle every units. But real-world signals oscillate at wildly different speeds — a heartbeat cycle lasts about seconds while an ocean wave might take seconds. The value controls this.
Past Knowledge
You know the period of is and can graph one cycle using five key points.
Today's Goal
Calculate the period from and graph waves with different periods.
Future Success
Period is the second transformation parameter. Combined with amplitude, you can model any wave height and speed.
Key Concepts
The Period Formula
For or :
Period
tells you how many cycles fit in radians.
Visual Comparison
The Effect of
| Condition | Period | Effect |
|---|---|---|
| Horizontal compression — faster oscillation | ||
| Standard sine/cosine wave | ||
| Horizontal stretch — slower oscillation |
A Common Source of Confusion
Bigger means a smaller period (faster oscillation). This is the opposite of amplitude, where bigger means a bigger wave. Think of as “how many cycles per ” — more cycles = shorter period.
Worked Examples
Finding the Period
Question: Find the period of .
Step 1: Identify .
Step 2: Apply the formula: Period
Final Answer: Period . Three complete cycles fit in radians.
Finding the Five Key Points
Question: List the five key points of for one cycle starting at .
Step 1: Period . Quarter-period .
Step 2: -values:
Step 3: Cosine pattern (max → zero → min → zero → max):
Final Answer: , , , ,
Finding B from the Period
Question: A sine wave completes one cycle every units. Write the equation.
Step 1: The period is . Set up the equation:
Step 2: Solve for :
Final Answer:
Common Pitfalls
Using as the Period
is NOT the period — it is the frequency factor. The period is . For , the period is , not .
Real-Life Applications
Musical Pitch
The note “middle C” vibrates at approximately Hz, meaning the air pressure variations complete full sinusoidal cycles per second. That directly corresponds to a value of in the equation . When a string vibrates at double the frequency ( Hz), the period halves — and you hear the note one octave higher.
Practice Quiz
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