Amplitude and Period

Learn how amplitude and period transform sine and cosine waves and control their height and width.

Advanced25 minLesson
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Definition

The amplitude and period are two fundamental properties that describe how a trigonometric function behaves.

Amplitude

The amplitude is the distance from the midline to the maximum (or minimum) of the wave. For the general form:
y=Asin⁡(Bx)ory=Acos⁡(Bx)y = A \sin(Bx) \quad \text{or} \quad y = A \cos(Bx)
The amplitude is ∣A∣|A| (the absolute value of AA).
  • When ∣A∣>1|A| > 1: The wave is stretched vertically (taller)
  • When 0<∣A∣<10 < |A| < 1: The wave is compressed vertically (shorter)
  • When A<0A < 0: The wave is reflected across the x-axis

Period

The period is the horizontal length of one complete cycle. For the general form:
y=Asin⁡(Bx)ory=Acos⁡(Bx)y = A \sin(Bx) \quad \text{or} \quad y = A \cos(Bx)
The period is 2π∣B∣\frac{2\pi}{|B|}.
  • When ∣B∣>1|B| > 1: The wave is compressed horizontally (more cycles in the same space)
  • When 0<∣B∣<10 < |B| < 1: The wave is stretched horizontally (fewer cycles)

Try it now

What is the amplitude of y=5sin⁡(x)y = 5\sin(x)?

Worked Examples

Find the amplitude of y=3sin⁡(x)y = 3\sin(x)

1

Identify the coefficient A

Comparing y=3sin⁡(x)y = 3\sin(x) with y=Asin⁡(Bx)y = A\sin(Bx) → A=3A = 3

2

Calculate amplitude

Amplitude =∣A∣=∣3∣= |A| = |3| → Amplitude=3\text{Amplitude} = 3

3

Interpret the result

The wave reaches 3 units above and below the midline → Maximum: 3, Minimum: −3-3

Common Mistakes

Thinking amplitude can be negative

Why it's wrong: Amplitude is always positive because it represents distance. When you see y=−3sin⁡(x)y = -3\sin(x), the amplitude is still 3.

Correct: Amplitude =∣A∣= |A|. A negative AA causes reflection, not negative amplitude.

Confusing BB with the period

Why it's wrong: Students often think BB is the period. Actually, BB affects how compressed the wave is.

Correct: Period =2π∣B∣= \frac{2\pi}{|B|}. As BB increases, period decreases (inverse relationship).

Forgetting the 2π2\pi in the period formula

Why it's wrong: The standard sine/cosine wave has period 2π2\pi, not 1. All period calculations are based on this.

Correct: Period =2π∣B∣= \frac{2\pi}{|B|}, not 1B\frac{1}{B}. The 2π2\pi comes from the natural period of sine and cosine.

Calculating period of sin⁡(x/2)\sin(x/2) as 2π1/2=π\frac{2\pi}{1/2} = \pi

Why it's wrong: Division by a fraction requires flipping and multiplying.

Correct: 2π1/2=2π×2=4π\frac{2\pi}{1/2} = 2\pi \times 2 = 4\pi. When B<1B < 1, the period is LONGER than 2π2\pi.

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Practice Problems

16 problems
Problem 1 of 16
Easy

What is the amplitude of y=5sin⁡(x)y = 5\sin(x)?

Why It Matters

Amplitude and period appear everywhere waves exist:
  • Sound waves: Amplitude determines volume (louder = larger amplitude), period determines pitch (higher pitch = shorter period)
  • Ocean waves: Amplitude is wave height, period is time between waves
  • Electricity: AC power has amplitude (voltage) and period (60 Hz in the US = period of 160\frac{1}{60} second)
  • Music: Every musical note is a wave with specific amplitude and period
  • Light: Different colors have different periods (wavelengths)
Engineers, musicians, and scientists use these concepts daily to design everything from speakers to bridges to MRI machines!

Real World Applications

Sound Engineering

Audio engineers adjust amplitude (volume) and period (pitch) to mix music and design speakers.

Example:

A bass drum produces waves with long period (low frequency ~60 Hz), while a cymbal has short period (high frequency ~10,000 Hz).

1Try It Yourself

A speaker produces a tone modeled by y=Asin⁡(440⋅2πt)y = A\sin(440 \cdot 2\pi t) where tt is in seconds. This is the note A4 (440 Hz).

What is the period of this wave in seconds?

Step 1: Write the mathematical expression

Use the period formula with B=440⋅2πB = 440 \cdot 2\pi:

Electricity and Power

AC (alternating current) electricity follows a sine wave pattern. In the US, the standard is 120V at 60 Hz.

Example:

The voltage can be modeled as V(t)=170sin⁡(120πt)V(t) = 170\sin(120\pi t). The amplitude is 170V (peak voltage), and the period is 160\frac{1}{60} second.

2Try It Yourself

European power uses 50 Hz instead of 60 Hz.

What is the period of European AC power?

Step 1: Write the mathematical expression

Calculate 1frequency\frac{1}{\text{frequency}}:

Key Takeaways

  • 1Amplitude =∣A∣= |A| determines the height of the wave (distance from midline to peak)
  • 2Period =2π∣B∣= \frac{2\pi}{|B|} determines the width of one complete cycle
  • 3Larger ∣A∣|A| makes the wave taller; smaller ∣A∣|A| makes it shorter
  • 4Larger ∣B∣|B| compresses the wave horizontally; smaller ∣B∣|B| stretches it
  • 5For y=Asin⁡(Bx)y = A\sin(Bx) or y=Acos⁡(Bx)y = A\cos(Bx): identify AA and BB, then apply the formulas

Frequently Asked Questions

If A=0A = 0, the function becomes y=0y = 0 (a flat horizontal line). There is no wave at all.
If A=0A = 0, the function becomes y=0y = 0 (a flat horizontal line). There is no wave at all.
No. Period is a distance (length of one cycle), so it is always positive. That's why we use ∣B∣|B| in the formula.
Since B=1B = 1, the period is 2π1=2π\frac{2\pi}{1} = 2\pi. This is the 'standard' period for sine and cosine.
Find the maximum value and minimum value, then: Amplitude =max−min2= \frac{\text{max} - \text{min}}{2}

Glossary

Amplitude
The maximum displacement from the midline of a wave; equals ∣A∣|A| in y=Asin⁡(Bx)y = A\sin(Bx)
Period
The horizontal length of one complete cycle; equals 2π∣B∣\frac{2\pi}{|B|} in y=Asin⁡(Bx)y = A\sin(Bx)
Frequency
The number of cycles per unit; frequency =1period= \frac{1}{\text{period}}
Midline
The horizontal line halfway between the maximum and minimum values of a wave
Cycle
One complete repetition of a periodic function's pattern

Formula Card

General Form

y=Asin⁡(Bx)y = A\sin(Bx) or y=Acos⁡(Bx)y = A\cos(Bx)

Standard form of sine and cosine with amplitude A and frequency B

Amplitude

∣A∣|A|

Height of wave from midline to peak

Period

2π∣B∣\frac{2\pi}{|B|}

Horizontal length of one complete cycle

Frequency

∣B∣2π\frac{|B|}{2\pi}

Number of cycles per unit

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